Katherine Stange

Hao Chen

Shalom Eliahou

Éric Colin

Maud Joubaud

Abel Lacabanne

Jérémy Nusa

The-Cang Nguyen

Francesco Bonaldi

Néstor Fernández Vargas

Pelle Steffens

Robert Rapadamnaba

Ibrahim Abdoulrahim

Emmanuel Youssouf

Thiziri Gnandi

Mario Moulla

Nicolás Veruete

Léon

Ruffini Berardo

Je Tiens

Sophie Cazanave

Carmela Madonia

Éric Hougounenq

Baptiste Cha- Puisat

Nathalie Collain

Christian Laura

Pelu Adro Patri

David

Iván González Tobar

Tertulia De Sobremesa

José Mi Familia Cacereña Irina

Nahuel

avec qu'on a partagé des bons moments ensembles

En premier lieu, je voudrais remercier mon directeur de thèse, Jorge Ramírez, pour m'avoir donné l'opportunité de travailler avec lui sur des sujets passionnants, toujours avec patience, bienveillance et enthusiasme, tout en me transmettant un regard esthétique des mathématiques. En effet, les mots que j'ai les plus entendu de lui lorsqu'une relation surprenante apparaissait sur son tableau n'étaient pas "C'est intéressant..." mais "¡Qué bonito!".

Ma reconnaissance va également aux membres du jury de ma thèse,

Résumé

Les empilements apolloniens ont attiré l'attention des mathématiciens en raison de leurs applications en théorie des nombres, théorie géométrique des groupes, géométrie hyperbolique, structures fractales et géométrie discrète. Dans cette thèse, nous introduisons une classe d'empilements de sphères où la combinatoire est donnée par un polytope inscrit aux arêtes. À travers cette connexion, nous généralisons les empilements apolloniens dans d'autres contextes géométriques et en dimensions supérieures. Cette structure polytopale permet également d'obtenir une généralisation du théorème de Descartes pour les empilements de sphères provenant des polytopes réguliers dans toutes les dimensions. Nous utilisons ce résultat pour caractériser l'intégralité des empilements apolloniens provenant des solides platoniciens. Puis, nous introduisons la notion de section apollonienne, et nous l'utilisons pour montrer que l'ensemble des courbures de tout empilement apollonien intégral tétraédrique, octaédrique ou cubique est contenu dans l'ensemble des courbures d'un empilement apollonien orthoplicial intégral. L'approche polytopale que nous proposons nous permet également d'étendre les applications des empilements apolloniens dans une nouvelle direction dans le domaine de la topologie. Dans cette thèse, nous introduisons deux méthodes de construction de représentations en colliers de noeuds et d'entrelacs. La première méthode découle directement du théorème d'empilements de cercles de Koebe-Andreev-Thurston, et donne une borne supérieure linéaire sur le nombre minimal de sphères nécessaires pour construire une représentation en collier en termes du nombre minimal de croisements. Dans la seconde méthode, nous utilisons la structure fractale des empilements apolloniens orthopliciaux pour construire des représentations en collier des entrelacs rationnels avec des propriétés arithmétiques intéressantes.

Résumé étendu

Les empilements de sphères sont des objets qui apparaissent très souvent, non seulement en mathématiques, mais aussi en divers champs scientifiques comme la chimie pour les structures cristallines, la physique pour la modélisation des milieux granulaires ou l'informatique pour les codes correcteurs. Parmi la variété infinie des configurations possibles, les empilements apolloniens ont attiré l'attention des mathématiciens en raison de leurs applications en théorie des nombres, théorie géométrique de groupes, géométrie hyperbolique, structures fractales ou géométrie discrète. Dans cette thèse, nous utilisons la combinatoire des polytopes pour généraliser les empilements apolloniens et les appliquer dans la construction polygonale des noeuds et entrelacs dans l'espace.

En premier lieu, nous introduisons les différents modèles géométriques qui sont utilisés pour étudier les empilements de sphères. Nous rappelons la bijection qui existe entre l'ensemble des boules euclidiennes en dimension d et l'ensemble des vecteurs unitaires dans l'espace lorentzien en dimension d + 2. Cette bijection fait de l'algèbre linéaire, appliquée aux espaces lorentziennes, un outil très adapté pour étudier les empilements de sphères.

Nous continuons avec les notions basiques de la théorie des noeuds, en faisant un point sur les entrelacs rationnels introduits par Conway, et finissons cette section en décrivant les invariants géométriques qui sont liés aux représentations polygonales des noeuds. La longueur en sphères est l'un de ces invariants, défini comme le nombre minimal de sphères dans un collier ayant la forme d'un noeud ou entrelacs donné. Dans le chapitre 2, en utilisant la géométrie lorentzienne et le théorème de Koebe-Andreev-Thurston sur les empilements de cercles, nous prouvons une borne linéaire de la longueur en sphères en fonction du nombre minimal de croisements d'un noeud. La méthode utilisée dans la preuve inspire un premier algorithme de construction des représentations de noeuds en collier à partir d'un diagramme de noeuds planaire. Ce type de représentations a été récemment utilisée pour l'étude du volume hyperbolique de 3-variété hyperbolique. L'application des empilements apolloniens dans la théorie des nombres vient du fait qu'il existe une infinité d'empilements apolloniens intégraux dont les courbures de tous les cercles contenus dans l'empilement sont des entiers. C'est le théorème de Descartes sur les courbures d'un empilement de 4 cercles deux-à-deux tangents qui permet de créer le lien, car il donne les conditions nécessaires pour qu'un empilement apollonien soit intégral. Dans le chapitre 4, nous utilisons la géométrie lorentzienne pour donner une généralisation du théorème de Descartes pour les empilements polytopaux réguliers. Nous appliquerons ce résultat pour caractériser l'intégralité des empilements apolloniens polytopaux, lorsque le polytope correspondant est l'un des solides de Platon.

Le chapitre 5 est dédié à l'étude d'une famille d'empilements polytopaux réguliers en dimension 3: les empilements orthopliciaux. Ceux-ci sont des empilements ayant la combinatoire de l'orthoplex, l'analogue de l'octaèdre en dimension 4. La version apollonienne pour les empilements orthopliciaux a été introduite par Dias et Nakamura en 2014, comme une généralisation des empilements apolloniens trois dimensionnelles ayant une structure algébrique plus riche. Cela leur a permis de résoudre des problèmes arithmétiques, dans les cas des empilements apolloniens orthopliciaux, qui restent toujours ouverts pour les empilements apolloniens classiques en dimension 2. Avec les outils développés dans les chapitres précédents,

Introduction

Sphere packings appear diversely in chemistry, physics, computer science and mathematics. There is a long trajectory on numerical and analytic approaches, but many questions about the algebraic and combinatorial aspects of non-congruent sphere packings remain open to date. Among the great variety of possible configurations, the Apollonian packings have drawn increasing attention by mathematicians due to their applications in number theory, geometric group theory, hyperbolic geometry, fractal structures or discrete geometry.

The origin of the term Apollonian packings goes back to a problem apparently raised by Apollonius of Perga around 230 BCE, following the writings of Pappus of Alexandria on Apollonius' works [START_REF] Ver Eecke | Pappus d'Alexandrie: la collection mathématique[END_REF]. The problem of Apollonius concerned the number of circles that can be added to be tangent to three given circles. If the three given circles are pairwise tangent, then the answer is two. The packing made by these five circles gives the first step towards an Apollonian packing, which is made by inscribing a new circle at the interstice made by each triplet of circles, and then repeating this process ad infinitum. The first record of the whole picture belongs to Leibniz, in a fragment of a letter to Des Bosses in 1706 [START_REF] Leibniz | Leibniz to Des Bosses: 11 March 1706[END_REF] on his attempts to illustrate the notion of infinitesimal.

Beyond mathematics, the space-filling property of the three-dimensional analogue of Apollonian packings has been used in the modelling of granular systems [START_REF] Anishchik | Three-dimensional Apollonian packing as a model for dense granular systems[END_REF], foam structures [START_REF] Sauerbrei | The Apollonian decay of beer foam bubble size distribution and the lattices of young diagrams and their correlated mixing functions[END_REF], fluid emulsions [START_REF] Kwok | Apollonian packing in polydisperse emulsions[END_REF] and in many other applications [START_REF] Varrato | Apollonian packings as physical fractals[END_REF][START_REF] Stäger | Cutting self-similar space-filling sphere packings[END_REF][START_REF] Kranz | Complex networks from space-filling bearings[END_REF].

In number theory, research on Apollonian packings has been increasing in the last decades, due to the interesting properties of the integral and algebraic structures behind Apollonian packings revealed in a foundational series of papers by Graham, Lagarias, Mallows, Wilks and Yan [START_REF] Graham | Apollonian circle packings: number theory[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory i. the Apollonian group[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory II. super-Apollonian group and integral packings[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory III. higher dimensions[END_REF]. Chen, in his PhD thesis [START_REF] Chen | Apollonian ball packings and stacked polytopes[END_REF],

proposed a combinatorial approach to study Apollonian packings by relating them with stacked polytopes. In this thesis, we develop a line of research close to Chen's, by using a polytopal approach to generalize the structure of Apollonian packings in different geometric settings and in higher dimensions.

Contents

The approach that we propose allowed us to obtain new results in number theory on Apollonian packings, and also to extend the applications of Apollonian packings into a novel direction in the area of topology. For a given knot K (resp. link), a necklace representation is a cyclic chain (resp. chains) of consecutive tangent spheres which are ambient isotopic to K. Necklace representations of knots and links have been recently used for the study of hyperbolic volume of hyperbolic 3-manifolds [START_REF] Gabai | Hyperbolic 3-manifolds of low cusp volume[END_REF]. In this thesis, we used the fractal structure of polytopal Apollonian packings to construct necklace representations of links and knots with interesting arithmetical properties.

Organization of the manuscript

In Chapter 1, we discuss the classic results and definitions in sphere packings and knots that will be used throughout the rest of the manuscript.

In Chapter 2, we present an upper bound on the minimum number of spheres needed to construct a necklace representation of a given knot or link. We finally describe an algorithm to construct necklace representations, which is based in the constructive proof of the upper bound.

In Chapter 3, we explore the connection between polytopes and sphere packings in any dimension by introducing a class of packings whose combinatorics is based on edge-scribed polytopes. We then study the unicity under Möbius transformations and the duality of these packings. The latter allows us to generalize the Apollonian group and the Apollonian packing within new settings.

In Chapter 4, we generalize the Descartes' theorem for polytopal d-ball packings obtained from a regular polytope. The latter allows us to construct integral Apollonian packings based on the Platonic solids.

In Chapter 5, we review the orthoplicial Apollonian packing, introduced by Dias [START_REF] Dias | The Local-Global Principle for integral generalized Apollonian sphere packings[END_REF] and Nakamura [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF], with the tools developed in the previous chapter. After introducing the notion of Apollonian section, we show that the set of curvatures of every integral tetrahedral, octahedral and cubical Apollonian packing is contained in the set of curvatures of an integral orthoplicial Apollonian packing.

The spherical model

Let d ≥ 1 be an integer. We denote R d the Euclidean space of dimension d. Let -Solid sphere: {x ∈ R d | ∥x -c∥ ≤ 1/κ} when κ > 0.

It is also a standard d-ball of R d with center c ∈ R d and radius 1 κ .

-Hollow sphere: {x ∈ R d | ∥x -c∥ ≥ -1/κ} when κ < 0.

It can be regarded as the closure of the complement of a solid sphere.

-Half-space: {x ∈ R d | x • n ≤ δ} when κ = 0.

By convention, we choose the normal vector n pointing towards the interior.

The real number δ represents the signed distance from the boundary to the origin (positive if the origin is contained in the interior and negative otherwise).

There is a natural embedding of Balls( R d ) → Balls( R d+1 ) where a d-ball b of center c and curvature κ (resp. normal vector n and signed distance δ) is mapped to a (d+1)ball b of center (c, 0) and curvature κ (resp. normal vector ( n, 0) and signed distance δ). We call this embedding the inflate operation. -

Orthogonal if ∡(b, b ′ ) = π 2 .
-Externally tangent if ∡(b, b ′ ) = π.

When the boundaries of b and b ′ do not intersect the intersection angle ∡(b, b ′ ) is not well-defined. In this case we say that b and b ′ are disjoint if they have disjoint interiors and nested if one is contained in the other.

Remark 1. The inflate operation preserves intersection angles.

We notice that the definition of intersection angle does not apply when d = 1, since the boundary of a 1-ball is not simply connected. In this case, we can define the intersection angle of two 1-balls as the intersection angle of the corresponding 2-balls given by the inflate operation. measured at a non-common region.

The hyperbolic model

The Lorentzian model

The Lorentzian space of dimension d + 2, denoted by L d+1,1 , is a real vector space of dimension d + 2 equipped with a bilinear symmetric form ⟨•, •⟩ of signature (d + 1, 1). The Lorentzian product of two vectors x and y of L d+1,1 is the real number ⟨x, y⟩ and the Gramian of a collection of vectors B = {x 1 , . . . , x n } of L d+1,1 is the matrix From now on, we fix an orthonormal basis B 0 which we shall refer to as the canonical basis of L d+1,1 . A vector x ∈ L d+1,1 is called:

Gram(B) :=     ⟨x 1 , x 1 ⟩ • • • ⟨x 1 , x n ⟩ . . . . . . . . . ⟨x n , x 1 ⟩ • • • ⟨x n , x n ⟩     If B = {x 1 , . . . , x d+2 } is a basis of L d+1
-Space-like if ⟨x, x⟩ > 0.

-Time-like if ⟨x, x⟩ < 0.

-Light-like if ⟨x, x⟩ = 0.

-Future-directed (resp. past-directed) if ⟨e d+2 , x⟩ > 0 (resp. < 0).

-Normalized if |⟨x, x⟩| = 1.
The space of all the normalized space-like (resp. time-like) vectors of L d+1,1 is usually called de Sitter space (resp. anti de Sitter space). We denote it by S(L d+1,1 ) (resp.

T(L d+1,1 )). The anti de Sitter space can be regarded as the generalization of a twosheets hyperboloid with two connected components T ↑ (L d+1,1 ) and T ↓ (L d+1,1 ) formed by the future-directed and the past-directed vectors of T(L d+1,1 ) respectively. The hyperboloid model of the (d + 1)-hyperbolic space is obtained by taking T ↑ (L d+1,1 ) with the metric induced by the restriction of the Lorentzian product of L d+1,1 . The isomorphism which maps the hyperboloid model to the Poincaré ball model can be regarded as the projection π : T ↑ (L d+1,1 ) → {e d+2 = 0} from -e d+2 (see Figure 1.

2).

A time-like half-space is a subset

t x = {y ∈ L d+1,1 | ⟨x, y⟩ ≥ 0} where x ∈ S(L d+1,1
). The set of time-like half-spaces is in bijection with S(L d+1,1 ). The image

π(t x ∩ T ↑ (L d+1,1
)) is a hyperbolic half-space of H d+1 and every hyperbolic half-space can be obtained in this way. We can then extend the isomorphisms of (1.2) by

Balls( R d ) Caps(S d ) Halfs(H d+1 ) S(L d+1,1 ) ≃ ≃ ≃ (1.4)
The Lorentzian vector of a d-ball b, denoted by x b , is the normalized space-like vector obtained by the previous isomorphisms.

′ ⟩ =        cosh d H (h b , h b ′ ) if b and b ′ are nested cos ∡(b, b ′ ) if ∂b and ∂b ′ intersect -cosh d H (h b , h b ′ )
⟩                  < -1

The Möbius group

The Möbius Group Möb( R d ) can be defined as the group of the continuous auto- The Möbius Group defines a group action (on the left) on the space of d-balls.

The following group isomorphisms and equivariant group actions can be obtained by using the isomorphisms given in (1.4)

Balls( R d ) Caps(S d ) Halfs(H d+1 ) S(L d+1,1 ) Möb( R d ) Möb(S d ) Isom(H d+1 ) O ↑ (L d+1,1 ) ≃ ≃ ≃ ≃ ≃ ≃ (1.6)
where Möb(S d ) is the Möbius group defined on S d acting on the family of d-spherical caps, Isom(H d+1 ) is the group of hyperbolic isometries acting on the space of hyperbolic half-spaces and O ↑ (L d+1,1 ) is the Orthochronous Lorentz Group which is the group of linear maps of L d+1,1 preserving the Lorentz product and the time orientation. The latter acts on the space of normalized space-like vectors of L d+1,1 .

Moreover, for any

b ∈ Balls( R d ), the isomorphism Möb( R d ) → O ↑ (L d+1,1
) maps the inversion s b to the Lorentzian reflection on the boundary of the time-like half-space t x b , which corresponds to the linear map

y → y -2⟨y, x b ⟩ x b (1.7)
Since the Orthochronous Lorentz Group preserves the Lorentz product, the Möbius Group preserves the inversive product of d-balls.

Coordinate systems

Based on the previous models, we shall describe two different systems of coordinates for the space of d-balls.

Inversive coordinates

In [START_REF] Wilker | Inversive geometry[END_REF], Wilker defined the inversive coordinates of a d-ball b as the column-matrix i(b) given by the Cartesian coordinates of the Lorentzian vector of b with respect to the canonical basis. We shall use these coordinates for most of the computations.

The inversive coordinates can be obtained in terms of the curvature κ and center c (normal vector n and signed distance δ for half-spaces) by

i(b) := c B 0 (x b ) =        (κc, 1 2 (κ∥c∥ 2 -κ -1 -κ), 1 2 (κ∥c∥ 2 -κ -1 + κ)) T if κ ̸ = 0, ( n, δ, δ) T if κ = 0 (1.8)
where ∥ • ∥ denotes the Euclidean norm. The curvature of b can be deduced from its inversive coordinates by

κ(b) = -⟨x N , x b ⟩ (1.9) = k d+2 i(b) (1.10)
where x N = e d+1 + e d+2 and e i denotes the i-th vector of B 0 and k d+2 is the rowmatrix (0, . . . , 0, -1, 1) of length d + 2.

Points of R d can be seen as d-balls of infinite curvature. By extending continuously the isomorphism between Balls( R d ) and S(L d+1,1 ) we have that points of R d 1.1. The space of oriented hyperspheres 13 correspond to asymptotic directions in the light cone L(L d+1,1 ) made by all the lightlike vectors of L d+1,1 . Each direction corresponds to a point in the projective space

PL(R d+1,1
). Therefore, we can assign (homogeneous) inversive coordinates to a point

η ∈ R d by i(η) =        (2η, ∥η∥ 2 -1, ∥η∥ 2 + 1) T if η ̸ = ∞ (0 d , 1, 1) if η = ∞ (1.11)
where 0 d is the null vector of length d. Homogeneity means that for every λ ̸ = 0, λi(η) are valid inversive coordinates of the same point of R d (see [START_REF] Wilker | Inversive geometry[END_REF] for more details).

The following formula to compute the inversive product in inversive coordinates follows from Eq. (1.3)

⟨b, b ′ ⟩ = i(b) T Q d+2 i(b ′ ) (1.12)
where Q d+2 = diag(1, . . . , 1, -1) is the Gramian of B 0 . Therefore, the Möbius group is isomorphic to the following group of matrices

O ↑ d+1,1 (R) = {M ∈ GL d+2 (R) | M T Q d+2 M = Q d+2 and M d+2,d+2 > 0} (1.13)
where the isomorphism is obtained by applying Eq. (1.7), which gives the map

s b → I d+2 -2i(b)i(b) T Q d+2 (1.14)
where s b is the inversion on the boundary of b ∈ Balls( R d ) and I d+2 is the identity matrix of size d + 2.

Polyspherical coordinates

These coordinates were introduced by Boyd in [START_REF] Boyd | A new class of infinite sphere packings[END_REF]. Let B = {x 1 , . . . , x d+2 } be a basis of L d+1,1 . The polyspherical coordinates of a vector x ∈ L d+1,1 with respect to B, are the column matrix

p B (y) = ⟨x 1 , y⟩ • • • ⟨x d+2 , y⟩ T (1.15)

Chapter 1. General background

The polyspherical and Cartesian coordinates with respect to B are related by

c B (x) = Gram(B) -1 p B (x) (1.16)
By combining Equations (1.3) and (1.16), we can compute the Lorentzian product in polyspherical coordinates by

⟨x, y⟩ = p B (x) T Gram(B) -1 p B (y) (1.17)
In practice, we shall use Eq. (1.17) to compute the inversive product in a different basis.

Remark 2.

In the literature, the most used system of coordinates is the augmented-curvaturecenter coordinates introduced by Graham et al. in [START_REF] Graham | Apollonian circle packings: Geometry and group theory i. the Apollonian group[END_REF]. This system of coordinates shall not be used in this manuscript.

Packings

We Möbius Group preserves tangency graphs and maps carriers to carriers [START_REF] Stephenson | Introduction to circle packing: The theory of discrete analytic functions[END_REF]. In this manuscript, we shall study d-ball packings for d = 2, 3. Following the tradition, we shall refer to them as circle and sphere packings, respectively. Circle packable graphs were fully characterized in 1936 by Koebe [START_REF] Koebe | Kontaktprobleme der konformen Abbildung[END_REF]. The latter was rediscovered by Thurston by using some results of Andreev on hyperbolic 3-polytopes.

The space of oriented hyperspheres

The well-known Koebe-Andreev-Thurston Circle Packing Theorem (KAT theorem) is stated below. For a detailed survey on the applications of the KAT theorem we refer the readers to a paper of Bowers [START_REF] Bowers | Combinatorics encoding geometry: the legacy of Bill Thurston in the story of one theorem[END_REF]. The KAT theorem implies that the family of d-ball packable graphs is fully characterized for d = 1, 2. Such characterization is still unknown nowadays when d ≥ 3.

Indeed, d-ball packable graphs are closely related to the (d -1)-ball packable graphs which can be made by (d -1)-balls of the same size. It has been proved that the recognition of the tangency graphs of circle packings made by equal disks (and more generally disks with a bounded ratio between all the curvatures) is NP-hard (see [START_REF] Breu | On the complexity of recognizing intersection and touching graphs of disks[END_REF][START_REF] Hlineny | Touching graphs of unit balls[END_REF]). However, many properties of sphere packable graphs have been found

(see [START_REF] Kuperberg | Average kissing numbers for noncongruent sphere packings[END_REF][START_REF] Maehara | On the graph represented by a family of solid balls on a table[END_REF][START_REF] Maehara | On knotted necklaces of pearls[END_REF][START_REF] Bezdek | Contact graphs of unit sphere packings revisited[END_REF][START_REF] Chen | Lorentzian Coxeter systems and Boyd-Maxwell ball packings[END_REF]).

G → [B G ] i j and ψ : B ′ G → [B ′ G ] i j be two standard transformations. (Sufficiency) If there is a Euclidean isometry γ : [B G ] i j → [B ′ G ] i j then ψ -1 • γ • ϕ defines a Möbius transformation mapping B G to B ′ G .
(Necessity) Let us suppose that there is a Möbius transformation µ :

B G → B ′ G . Then θ := ψ • µ • ϕ -1 is a Möbius transformation mapping [B G ] i j to [B ′ G ] i j
and leaving fixed the half-spaces b i and b j . Therefore, θ is generated by inversions on d-balls which are simultaneously orthogonal to b i and b j . A d-ball simultaneously orthogonal to two parallel half-spaces must also be a half-space. Therefore, θ can be expressed as a product of Euclidean reflections so θ is a Euclidean isometry.

Preliminaries on knot theory

The main results and notations were adapted from the classic books of Adams [START_REF] Adams | The knot book[END_REF] and Cromwell [START_REF] Cromwell | Knots and Links[END_REF].

Diagrams of knots and links

A link L with n components consists of n disjoint simple closed curves in R 3 (or S 3 ). A knot K is a link with one component. Two links are said to be equivalent if there is an ambient isotopy of R 3 which carries one to the other. A link diagram L is a regular projection of a link L onto a plane in such a way that the projection of each component is smooth and at most two curves intersect at any point. At each crossing point of the link diagram, the curve which goes under the other is specified by deleting a small neighbourhood as in Figure 1.5. The crossing number of a L, denoted by cr(L), is the minimum number of crossings among all the diagrams of links which are equivalent to L. A link L is said to be:

Chapter 1. General background

(i) alternating if it admits an alternating diagram.

(ii) trivial if it is equivalent to a link admitting a diagram without crossings. Otherwise, L is said to be nontrivial.

(iii) splittable if a plane can be embedded in R 3 such that it separates one or more components of L. Otherwise, L is nonsplittable.

(iv) amphichiral if L is equivalent to its mirror, i.e. the link obtained from L after applying a reflection. Otherwise, L is chiral.

The main result concerning the crossing number of alternating links was conjectured by Tait [START_REF] Tait | On knots I[END_REF] and proved independently by Kauffman [START_REF] Kauffman | State models and the Jones polynomial[END_REF], Murasugi [START_REF] Murasugi | Jones polynomials and classical conjectures in knot theory[END_REF] and Thistlethwaite [START_REF] Thistlethwaite | A spanning tree expansion of the Jones polynomial[END_REF].

Theorem 1.2.1 (Tait's conjecture on the crossing number). Reduced alternating diagrams have the minimum number of crossings.

Preliminaries on knot theory 19

Tangles

A n-tangle is a pair (U, t) where U is a compact set of R 3 homeomorphic to a 3ball and t is a collection {γ 1 , γ 2 , . . . , γ m } of 1 ≤ n ≤ m disjoint arcs contained in U satisfying the following conditions:

(1) For every

1 ≤ i ≤ m, γ i is closed if and only if i > n.
(2) The endpoints of γ i for every 1 ≤ i ≤ n lie on ∂U (these are called the endpoints of the tangle).

Two n-tangles (U, t) and (U ′ , t ′ ) are said to be equivalent if there is an isotopy of R 3 carrying U to U ′ , t to t ′ and the endpoints of (U, t) to the endpoints of (U ′ , t ′ ). We shall denote this equivalence relation t ≃ t ′ . Up to equivalence, we may consider that the endpoints of t lie on a same plane H. A tangle diagram of (U, t) is a regular projection of t on H, together with U ∩ H and the crossing information (see Fig. 1.6).

The equivalence of tangles corresponds to the equivalence of their tangle diagrams via Reidemeister's moves [START_REF] Cromwell | Knots and Links[END_REF]. In order to simplify the notation, we shall refer to a tangle (U, t) by t when there is no ambiguity. From now on, we shall focus our attention on 2-tangles and their derived constructions. We shall name the endpoints in a 2-tangle diagram by the cardinal points NE, NW, SE and SW.

Chapter 1. General background

Let us now recall some classic operations of 2-tangles. For any 2-tangles t and t ′ we have the following binary operation (i) the sum t + t ′ , obtained by connecting the East endpoints of t to the West end- (ii) the mirror -t is the image of t under the reflection on the plane containing the equator, (iii) the flip F (t) is the image of t under the reflection on the plane perpendicular to the equator and passing through the endpoints SW and NE, (iv) the positive half-twist

points of t ′ , t ′ t t + t ′
H + : t → t 1 + t, (v) the negative half-twist H -: t → -t 1 + t. t -t F (t) H + (t) H -(t)

Rational tangles

Rational tangles were introduced by Conway in his work on enumerating and classifying knots and links [START_REF] Conway | An enumeration of knots and links, and some of their algebraic properties[END_REF]. For a given sequence of integers a 1 , . . . , a n all nonzero except maybe a 1 , we denote by t(a 1 , • • • , a n ) the rational tangle given as follows by Conway's algorithm

t(a 1 , • • • , a n ) := M (a 1 ) • • • • • M (a n )(t ∞ ) where M (a i ) := (H + ) a i F if a i ≥ 0 (H -) -a i F if a i < 0 (1.18)
An example of a rational tangle is illustrated in Fig. 1.11. The fraction of a rational tangle t(a 1 , . . . , a n ) is the continued fraction

M (-3) t ∞ M (-2) t(-3) M (2) t(-2, -3) t(2, -2, -3)
p q := a 1 + 1 a 2 + 1 . . .+ 1 an (1.19)

Chapter 1. General background

In Conway's famous paper on tangles [START_REF] Conway | An enumeration of knots and links, and some of their algebraic properties[END_REF], he gave a full characterization of rational tangles.

Theorem 1.2.2 (Conway). Two rational tangles t(a 1 , . . . , a n ) and t(b 1 , . . . , b m ) are equivalent if and only if

a 1 + 1 a 2 + 1 . . .+ 1 an = b 1 + 1 b 2 + 1 . . .+ 1 bm
The origin of the name of rational tangle came from the connection established by Conway's theorem, between the family of tangles produced by the Conways's algorithm and the rational numbers. We denote by t p/q the class of the rational tangles with fraction p/q under tangle equivalence. When all a i have the same sign in t(a 1 , . . . , a n ), then the corresponding tangle diagram is alternating. Since there is always a continued fraction expansion of p/q in which all the coefficients have the same sign [START_REF] Cromwell | Knots and Links[END_REF], then t p/q always has an alternating tangle diagram. Following Conway's definition, a rational link is the closure of a rational tangle.

Rational links are also the 2-bridge links, i.e. links admitting a representation such that the natural height function given by the z-coordinate has only two maxima and two minima as critical points [START_REF] Cromwell | Knots and Links[END_REF]. We shall denote by C(a 1 , . . . , a n ) the rational link obtained by the closure of t(a 1 , . . . , a n ) (numerator closure if a 1 ̸ = 0 and denominator closure otherwise). By the above discussion, rational links are alternating.

Moreover, Theorem 1.2.1 gives

a i > 0, for every 1 ≤ i ≤ n ⇒ cr(C(a 1 , . . . , a n )) = a 1 + . . . + a n (1.20)
Tangles which are obtained by sums and flips of rational tangles are called algebraic tangles [START_REF] Adams | The knot book[END_REF]. Equivalently, links which are obtained by the closure of algebraic tangles are said to be algebraic or arborescent [START_REF] Gabai | Genera of Arborescent Links[END_REF]. A Pretzel link (or knot) P (a 1 , . . . , a n ) is a particular case of algebraic link defined by

P (a 1 , . . . , a n ) := N (F (t(a 1 )) + • • • + F (t(a n ))) (1.21)
We show in Figure 1.12 the Pretzel knot P (3, -2, 3) which corresponds to the knot Figure 1.12: The Pretzel knot P (3, -2, 3).

Braids

We consider oriented tangles by giving an orientation to the arcs. The endpoints of an oriented tangle (U, t) shall be called inputs or outputs according to the orientations. By convention, we shall represent U by a rectangle in the tangle diagrams,

where the orientation will be read from the bottom to the top. An oriented n-tangle (U, t) is monotone if every arc of t never goes back. Thus, a monotone n-tangle does not have closed arcs.

Definition 1.2.1. A braid of n strands (or n-braid) is a monotone-oriented n-tangle.

The closure of an n-braid is defined by labelling the inputs and the outputs in linear ordering from 1 to n, and then connecting outputs and inputs with the same label by unlinked oriented arcs (Fig. 1.13). One might ask if every link can be represented as the closure of a n-braid. The well-known Alexander's Theorem gives a positive answer [START_REF] Alexander | A lemma on systems of knotted curves[END_REF].

Theorem 1.2.3 (Alexander). For any link L there is a closed n-braid equivalent to L.

The minimum number n satisfying the Alexander's Theorem is called the braid index of L, and the minimum number of crossings among all the braids representing L is called the braid length [START_REF] Van Cott | Relationships between braid length and the number of braid strands[END_REF]. We shall represent closed n-braids by diagrams in a square-grid with the crossings lying in the interior of the squares and we shall call these diagrams square-grid diagrams (Fig. 1.13). It is easy to see that every closed braid admits a square-grid diagram.

Polygonal representations

A natural way to better understand knots and links is by representing them as closed polygonal curves in the space. Polygonal representations of links have been of great interest not only in mathematics, but also in chemistry and physics in the study of molecular chains such as DNA and proteins [START_REF] Elrifai | On stick number of knots and links[END_REF][START_REF] Murasugi | Knot theory and its applications[END_REF]. Moreover, by adding more restrictions to polygonal representations, we can endow more geometric structures to knots. Lattice representations are a particular case of polygonal representations where all the vertices have integer coordinates in the space. These representations have been useful to study geometric knot invariants such as the rope length, defined as the minimal length needed of a rope with a fixed thickness in order to form a given knot [START_REF] Hong | Minimum lattice length and ropelength of knots[END_REF]. We define a necklace representation of a link as a polygonal representation which is contained in the carrier of a sphere packing. We show in Necklace representations of links have been appearing recently in the study of the hyperbolic volume of hyperbolic 3-manifolds [START_REF] Gabai | Hyperbolic 3-manifolds of low cusp volume[END_REF]. There are also other geometric knot invariants, which are defined for minimal polygonal representations.

The stick number and the lattice stick number of a link L are defined as the minimum number of segments needed to construct a polygonal and lattice representation of L, respectively. In [START_REF] Maehara | On configurations of solid balls in 3-space: Chromatic numbers and knotted cycles[END_REF], Maehara introduced the ball number of a link L, denoted by ball(L), as the minimum number of spheres needed to construct a necklace representation of L. Little is known about the behavior of the ball number with respect to other geometric invariants. For instance, the stick number of the trefoil knot and the Hopf link is known to be 6 in both cases (see [START_REF] Adams | Why knot: knots, molecules and stick numbers[END_REF] for a nice proof for the trefoil), while their lattice stick number is 12 and 8, respectively [START_REF] Huh | Lattice stick numbers of small knots[END_REF][START_REF] Hong | Links with small lattice stick numbers[END_REF]. By contrast, Maehara proved that the ball number of the Hopf link is 8, but the ball number of the trefoil is, nowadays, not known. Maehara conjectured that ball(3 1 ) = 12, and he and Oshiro showed that 9 ≤ ball(3 1 ) ≤ 12 [START_REF] Maehara | On knotted necklaces of pearls[END_REF][START_REF] Maehara | On configurations of solid balls in 3-space: Chromatic numbers and knotted cycles[END_REF]. As far as we are aware, these are the only known results concerning the ball numbers of links. We show in Fig. 1.15 a minimal polygonal, lattice and necklace representations of the trefoil that are known. Linear upper bounds on the stick and the lattice stick number in terms of the crossing number have been found [START_REF] Hong | Upper bound on lattice stick number of knots[END_REF][START_REF] Negami | Ramsey theorems for knots, links and spatial graphs[END_REF][START_REF] Calvo | Geometric knot spaces and polygonal isotopy[END_REF][START_REF] Huh | An upper bound on stick number of knots[END_REF] Given the connection of the ball number with KAT Theorem, a linear bound seems inevitable. The main difficulty in the method described above is to show that two spheres are enough to form the bridges, independently of the chosen link.

From links to circle packable graphs

Every link diagram L with at least one crossing leads to a 4-regular planar graph G L where the vertices are the crossings and the edges are the arcs joining the crossings. This graph is not simple in general (see Fig. Let us suppose that G L is connected and it has no loops. Since G L is 4-regular, the subgraph of P (L) induced by a crossing vertex and its 4 neighbors is the 1-skeleton of a square-pyramid. We shall call this graph the square-pyramidal graph. In the case considered, P (L) can be obtained as the union of n square-pyramidal graphs, where n is the number of crossings of L. Therefore, we have

|V (P (L))| = |V × | + |V m | = n + 1 2 (4n) = 3n (2.1)

Square-pyramidal disk arrangements

Square-pyramidal graphs are one of the building blocks for constructing the desired necklace representations. From now on, we shall consider square-pyramidal graphs ⊠ with the labelling on the vertices depicted in Figure 2.2.

x

-1 -2 1 2 Figure 2.2:
The labelling of a square-pyramidal graph ⊠.

We say that a circle packing is square-pyramidal if its tangency graph is the squarepyramidal graph. As we show below, square-pyramidal graphs are not Möbius unique, that is, there are square-pyramidal circle packings which are not Möbius equivalent. Therefore, all the properties of square-pyramidal circle packings and the added structures must be carefully verified in each equivalence class under Möbius transformations. In the following proposition, we describe the Moduli space M ⊠ of all the square-pyramidal circle packings under Möbius transformations.

Chapter 2. An upper bound on the ball number

We have that 1 < κ 1 < 4. Indeed, when κ 1 < 1 (resp. κ 1 > 4), the disks d 1 and d -1 (resp. d 2 and d -2 ) overlap and, when κ 1 = 1 (resp. 4), d 1 and d -1 (resp. d 2 and d -2 ) are tangent so the tangency graph would be other than ⊠ (see Figure 2.3). We notice that the collection of circle packings {[D ⊠ ] -1

x (κ 1 )} 1<κ 1 <4 are Euclidean non-congruent. Therefore, by Lemma 1.1.1, they represent different equivalence classes in M ⊠ . Moreover, these are the only possible standard square-pyramidal circle packings under Euclidean isometries. Hence, M ⊠ is in bijection to the open interval [START_REF] Adams | The knot book[END_REF][START_REF] Anishchik | Three-dimensional Apollonian packing as a model for dense granular systems[END_REF]. We define, for every i = 1, 2, -1, -2, the standard curvatures of a square-pyramidal circle packing D ⊠ as the real numbers 1 < κ i < 4 corresponding to the curvature of the disk

d -1 d x d 1 d 2 d -2 κ 1 = 1 d -1 d x d 1 d 2 d -2 κ 1 = 4 d -1 d x d 1 d 2 d -2 1 < κ 1 < 2 d -1 d x d 1 d 2 d -2 κ 1 = 2 d -1 d x d 1 d 2 d -2 2 < κ 1 < 4
d i ∈ [D ⊠ ] -i
x . The standard curvatures can be used to identify the equivalence class of D ⊠ in M ⊠ . We define also the smaller standard curvature

κ := min{κ 1 , κ 2 , κ -1 , κ -2 }
We define a square-pyramidal disk arrangement as the collection of disks (D ⊠ , d It is clear that the previous arguments work for any standard [D ⊠ ] -1

x (κ 1 ) with 1 < κ 1 < 4. Since the conditions defining the mirror disks and the tangency disks are preserved under Möbius transformations, the Lemma is also true for the class of

[D ⊠ ] -1
x (κ 1 ) in M ⊠ , for any 1 < κ 1 < 4. As is shown in the proof of Proposition 2.3.1, the union of all the classes contains all the square-pyramidal circle packings.

Chapter 2. An upper bound on the ball number

d 1 d 2 d -1 d -2 d x d * 1 d * 2 d t Curvature Center / n Inversive coordinates d x 0 (δ = 1) 0 -1 0 -1 1 1 d 1 κ 1 0 1 κ 1 -1 0 1 -κ 1 -1 κ 1 -1 d 2 1 2 √ κ 1 0 2 √ κ 1 0 2 κ 1 -1 2 κ 1 d -1 0 (δ = 1) 0 1 0 1 1 1 d -2 1 -2 √ κ 1 0 -2 -2 √ κ 1 0 2 κ 1 -1 2 κ 1 d * 1 √ κ 1 2 0 -1 0 - √ κ 1 2 -1 √ κ 1 κ 1 -2 2 √ κ 1 d * 2 0 (δ = 0) 1 0 1 0 0 0 d t -κ 1 √ κ 2 1 +4 0 2 κ 1 0 -2 √ κ 2 1 +4 κ 1 √ κ 2 1 +4 0 Figure 2.4: (Left)
The square-pyramidal disk arrangement containing the stan-

dard [D ⊠ ] -1
x (κ 1 ) with the center of d 1 lying on the y-axis. (Right) Curvature, center and inversive coordinates of the disk in terms of the curvature of d 1 .

Lemma 2.3.2. The following relations hold for every square-pyramidal disk-arrangement

(D ⊠ , d * 1 , d * 2 , d t )
and for every i = 1, 2:

(a) ⟨d i , d -i ⟩ = -1 -2κ i = -1 -8 κ j with i ̸ = j. (b) κ i = κ -i . (c) κ 1 κ 2 = 4. (d) -7 < ⟨d i , d -i ⟩ < -1. (e) (1 -⟨d 1 , d -1 ⟩)(1 -⟨d 2 , d -2 ⟩) = 16. (f) ∂d t ⊂ d 1 ∪ d 2 ∪ d -1 ∪ d -2 . (g) d * 1 , d * 2 and d t are mutually orthogonal. (h) s d * i (d j ) = d -j if i = |j| d j otherwise for every j ∈ {1, 2, -1, -2, t}.
Proof. The relations can be obtained by simple calculations by using the inversive coordinates given in Fig.

= 2 when κ 1 = κ 2 . d 1 d 2 d -1 d -2 dx d -1 d -2 d 1 d 2 d x d 1 d 2 d -1 d -2 d x Figure 2.5:
The closest disjoint pairs in darker gray in three different cases:

(left) κ = κ 1 = 1.33 (center) κ = κ 1 = κ 2 = 2 (right) κ = κ 2 = 1.6.

The crossing sphere arrangement

We define a square-pyramidal sphere packing B ⊠ as a sphere packing obtained by inflating a square-pyramidal circle packing. We define equivalently the closest and farthest disjoint pairs as in the planar case. Let (D ⊠ , d * 1 , d * 2 , d t ) be a square-pyramidal disk arrangement. We define, for every ε ∈ {+, -}, the crossing sphere arrangement We also define the crossing region X of a crossing sphere arrangement as

(B ⊠ , b * 1 , b * 2 , b t , b ε3 , b ′ ε3 )
X :=   b∈B ⊠ -b   ∩ b t .
We show in Figure 2.6 three examples of crossing sphere arrangements with their corresponding crossing regions.

b 1 b -2 b -1 b 2 X b 3 b ′ 3 b -1 b -2 b 1 b 2 X b ′ -3 b -3 b 1 b 2 b -1 b -2 X b ′ 3 b 3
Gram(B) -1 = 1 2          κ 2 -1 κ 2 -1 √ κ 0 -1 0 -1 0 0 κ 2 -1 -1 κ 2 √ κ 0 √ κ 0 √ κ 2 0 0 0 0 0 2          (2.2)
In order to show that the bridge-spheres are well-defined, we compute the polyspherical coordinates of b ε3 with respect to B using the definition of b ε3 and Eq.(1.5) which gives

p B (b ε3 ) = -1 -1 -1 1 λ z,3 T with λ z,3 ≥ 1 (2.3)
By using Eq. (1.17), we can normalize to get

λ z,3 = 3 + 2 √ κ -κ > 1 for every 1 < κ ≤ 2.
The latter insures, for every square-pyramidal sphere packing, the existence and the uniqueness of b ε3 , and hence of b

′ ε3 := s b * c (b 3 ). Moreover, ⟨b ε3 , b ′ ε3 ⟩ = ⟨b ε3 , s b * c (b ε3 )⟩ = ⟨b ε3 , b ε3 -2⟨b ε3 , b * c ⟩b * c ⟩ by (1.7) = 1 -2⟨b ε3 , b * c ⟩ 2 = -1 so b ε3 and b ′
ε3 are externally tangent. A sphere b is contained in the crossing region of the crossing sphere arrangement

(B ⊠ , b * 1 , b * 2 , b t , b ε3 , b ′ ε3 ) if and only if ⟨b i , b⟩ ≤ -1 for every b i ∈ {b x , b c , b f , b -c , b -f } and ⟨b t , b⟩ ≥ 1 (2.4)
By combining the invariance of the inversive product under inversions and the inflate operation, the intersection angles between the disks of a square-pyramidal disk arrangement and the mirror disks given in Lemma 2.3.2 (h), and the tangency conditions in the definition of b ε3 we obtain

⟨b x , b ′ ε3 ⟩ =⟨s b * c (b x ), s b * c (b ′ ε3 )⟩ = ⟨b x , b ε3 ⟩ = -1 ⟨b -c , b ′ ε3 ⟩ =⟨s b * c (b -c ), s b * c (b ′ ε3 )⟩ = ⟨b c , b ε3 ⟩ = -1 ⟨b f , b ′ ε3 ⟩ =⟨s b * c (b f ), s b * c (b ′ ε3 )⟩ = ⟨b f , b ε3 ⟩ = -1
For the rest of inversive products we use Lemma 2.3.2 (h), Eq. (1.17) and the inversive coordinates given in Fig. 2.4.

⟨b c , b ′ ε3 ⟩ = ⟨s b * c (b c ), s b * c (b ′ ε3 )⟩ = ⟨b -c , b ε3 ⟩ = p B (b -c ) T Gram(B) -1 p B (b ε3 ) = -1 -2κ + 1 -1 - √ κ 0 Gram(B) -1          -1 -1 -1 1 3 + 2 √ κ -κ          = -1 -2 √ κ < -1 for 1 < κ ≤ 2
By the same procedure, we find that for all 1 < κ ≤ 2 we have

⟨b -f , b ′ ε3 ⟩ =⟨s b * c (b -f ), s b * c (b ′ ε3 )⟩ = ⟨b -f , b ε3 ⟩ = 3 - 4 √ κ - 8 κ < -1 ⟨b t , b ′ ε3 ⟩ =⟨s b * c (b t ), s b * c (b ′ ε3 )⟩ = ⟨b t , b ε3 ⟩ = 2 + 2 √ κ -κ √ 4 + κ 2 ≥ 1
We now have all the ingredients to proceed with the proof of the upper bound given in Theorem 2.1.1.

The proof of the upper bound

Proof of Theorem 2.1.1. Let L be a nonsplittable and nontrivial link, and let L be a minimal crossing diagram of L. The two conditions on the link L imply that the patchwork graph P (L) = (V × ∪V m , E) has at least two crossing vertices. On the other hand, the minimality of the diagram L implies that P (L) is a simple planar graph.

Therefore, by the KAT theorem, there is a circle packing D P (L) whose tangency graph is P (L). Let B P (L) be the inflating of D P (L) . For every crossing vertex x ∈ V × , D P (L) admits a square-pyramidal disk arrangement with circle packing D ⊠ (x). Therefore, B P (L) admits a crossing sphere arrangement with square-pyramidal sphere packing B ⊠ (x) and bridge spheres {b εx3 , b ′ εx3 }. We notice that

D P (L) = x∈V × D ⊠ (x) and B P (L) = x∈V × B ⊠ (x).
We choose ε x such that the thread of the chain made by the spheres

(b c , b ε3 , b ′ ε3 , b -c ) is over/under the thread of the chain (b f , b x , b -f ) according to the diagram L. Let B (L)
be the arrangement of spheres made by the union of all the bridge spheres with the appropriate signs with respect to L for each crossing vertex, and let B L := B P (L) ∪ B (L) . If B L were a packing, then its carrier would contain a polygonal link ambient isotopic to L (by construction). Moreover, the number of spheres

|B L | = |B P (L) | + |B (L) | = 3cr(L) + 2cr(L) = 5cr(L) since L is a minimal crossing diagram.
It remains to show that B L is a packing. This is equivalent to showing the following three claims:

(1) B P (L) is a packing.

(2) Every sphere of B P (L) is at most tangent to every sphere of B (L) .

(3) B (L) is a packing.

Claim (1)] This is obtained directly by the fact that the inflating operation preserves the inversive product. Claim (2)] Let x be a crossing vertex with corresponding square-pyramidal disk ar-

rangement (D ⊠ , d * 1 , d * 2 , d t ), crossing sphere arrangement (B ⊠ , b * 1 , b * 2 , b t , b εx3 , b ′ εx3 ) and crossing region X . Since D P (L) is a packing then, as a consequence of Lemma 2.3.2 (f), any disk d ∈ D P (L) \ D ⊠ must be disjoint to d t . Therefore,
⊠ = {d x , d 1 , d 2 , d -1 , d -2 } and D ′ ⊠ = {d x ′ , d 1 ′ , d 2 ′ , d -1 ′ , d -2 ′ } be the corresponding square-pyramidal circle packings in D P (L) .
Let n be the number of disks in common of D ⊠ and D ′ ⊠ and let X and X ′ be the corresponding crossing regions. Now we show that in each of the five cases (n = 0, 1, 2, 3, 4), X and X ′ have disjoint interiors. We may relabel D ′ ⊠ in order to work with the labelling of the graphs shown on the left.

(n = 0) x 1 2 -1 -2 x ′ -2 ′ -1 ′ 2 ′ 1 ′
Since D P (L) is a packing then, by Lemma 2.3.2 (f), the boundaries of d t and d t ′ do not intersect.

Therefore, b t and b t ′ are disjoint. Hence, X ∩X ′ = ∅.

(n = 1) x

1 = 1 ′ 2 -1 -2 x ′ 2 ′ -2 ′ -1 ′
The (possibly empty) region

d t ∩ d t ′ must be con- tained in d 1 so b t ∩ b t ′ is contained in b 1 . As a con- sequence, int(X ) ∩ int(X ′ ) = ∅. (n = 2) x 1 = 1 ′ 2 = 2 ′ -1 -2 x ′ -2 ′ -1 ′
We can apply a standard transformation to get a

standard circle packing [D ⊠ ∪ D ′ ⊠ ] 1 2
where the disks d 1 , d 2 , d t and d t ′ become half-spaces as in Fig. 2.7.

d 1 d 2 d -1 d -2 d x d t d -1 ′ d -2 ′ d x ′ d t ′ d 1 d 2 d -1 d -2 d x d -1 ′ d -2 ′ d x ′ d t d t ′ Figure 2.7: (Left) D ⊠ ∪ D ′ ⊠ in the case n = 2 together with their tangency disks; (right)[D ⊠ ∪ D ′ ⊠ ] 1 2 .
The lines ∂d t and ∂d

t ′ in [D ⊠ ∪ D ′ ⊠ ] 1 2 either intersect in a point lying in d 1 ∪ d 2 or they are parallel implying, in both cases, that the region d t ∩ d t ′ is contained in d 1 ∪ d 2 . Therefore, b t ∩ b t ′ is contained in b 1 ∪ b 2 and thus int(X ) ∩ int(X ′ ) = ∅. (n = 3) x 1 = 1 ′ 2 = 2 ′ -1 = -1 ′ -2 x ′ -2 ′
The boundaries of d t and d t ′ intersect at the tan-

gency points d 1 ∩ d 2 and d -1 ∩ d 2 (Fig. 2.8). There- fore d t ∩ d ′ t ⊂ d 2 which implies that b t ∩ b ′ t ⊂ b 2 and hence int(X ) ∩ int(X ′ ) = ∅. d 1 d 2 d -1 d -2 dx d t d -2 ′ d x ′ d t ′ d 1 d 2 d -1 d -2 d x d -2 ′ d x ′ d t d t ′ Figure 2.8: (Left) D ⊠ ∪ D ′ ⊠ for n = 3 and the tangency disks (right)[D ⊠ ∪ D ′ ⊠ ] 1 2 .
(n = 4)

x 1 = 1 ′ 2 = 2 ′ -1 = -1 ′ -2 = -2 ′ x ′
In this case, the tangency graph of D ⊠ ∪ D ′ ⊠ is isomorphic to the octahedral graph by taking x ′ = 3 and x = -3 (Fig. 2.9). We have that

d t = -d t ′
which implies that b t and b t ′ are externally tangent.

Thus, int(X ) ∩ int(X ′ ) = ∅. d 1 d 2 d -1 d -2 d x d t d t ′ d x ′ d 1 d 2 d -1 d -2 d x d x ′ d t d t ′ Figure 2.9: (Left) D ⊠ ∪ D ′ ⊠ for n = 4 and the tangency disks (right)[D ⊠ ∪ D ′ ⊠ ] 1 2 .
Finally, since by Lemma 2.4.1 the bridge spheres are contained in the crossing regions we have that B (L) , and by consequence B L , are packings.

Remark 3. The method in the previous proof requires two spheres for each bridge in order to connect the closest pair. Unfortunately, this cannot be done with a single sphere since it would be too large to be contained in the crossing region, a central request in the proof.

An algorithm for knotted necklaces

Here we present an algorithm arising from the constructive proof of Theorem 2.1.1. We believe that this algorithm can be useful to investigate 3D invariants of links, such as the rope length or the stick number.

The spheres are given in inversive coordinates. To compute the bridge spheres we use the inversive products λ -c,3 Table 2.1: Algorithm to construct necklace representations.

:= ⟨b -c , b ε3 ⟩ = ⟨b c , b ′ ε3 ⟩ and λ z,3 := ⟨b εz , b ε3 ⟩ = ⟨b εz , b ′ ε3 ⟩.
Input: A link diagram L with n crossings of a link L.

Output: A necklace representation B L of the link L with 5n spheres.

1. Compute the patchwork graph P (L) of L.

2. With a Circle Packing algorithm, construct a circle packing D P (L) with tangency graph P (L).

3. Construct a sphere packing B P (L) obtained by inflating D P (L) .

Set B

(L) = {}, Q = diag(1, 1, 1, 1, -1), b z = ( 0 0 1 0 0 ) T .
5. For each crossing vertex x of P (L) do:

(a) Give to B ⊠ (x) a square-pyramidal labeling B ⊠ (x) = {b x , b 1 , b 2 , b -1 , b -2 }. (b) Compute the inversive product λ = b T 1 Qb -1 . (c) If λ ≥ -3, then B = (b x |b 1 |b 2 |b -1 |b z ) and κ = 1-λ 2 . Else, B = (b x |b 2 |b 1 |b -2 |b z ) and κ = 8 1-λ . (d) Define λ -c,3 = -1 -2 √ κ and λ z,3 = 3 + 2 √ κ -κ. (e) Set b 3 (x) = ( -1 -1 -1 λ -c,3 λ z,3 )B -1 Q) T and b ′ 3 (x) = ( -1 λ -c,3 -1 -1 λ z,3 )B -1 Q) T . (f)
′ 3 (x). (g) B (L) ← B (L) ∪ {b 3 (x), b ′ 3 (x)}. 6. B L := B P (L) ∪ B (L) .

Introduction

In this chapter, we study a class of packings which we call polytopal d-ball packings. These are packings arising from edge-scribed (d+1)-polytopes and are a central object of this thesis. One of the main differences is that the combinatorial structure of a polytopal d-ball packing is encoded by a polytope, instead of a graph, which is the case for general packings. Consequently, polytopal d-ball packings behave well under duality, and offer a good frame to generalize further constructions such as the Apollonian group and packings in higher dimensions.

As we will see, polytopal Apollonian packings encompass other known generalizations. For instance, the classic Apollonian Circle Packing, the generalization of Guettler and Mallows [START_REF] Guettler | A generalization of Apollonian packing of circles[END_REF] and the Apollonian ring packing with n = 5 [START_REF] Bolt | Apollonian Ring Packings. Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham[END_REF] are, respectively, a tetrahedral, octahedral and icosahedral Apollonian packing (Fig. 3.1). Even if the connection between polytopes and packings has drawn increasing attention in the last decades, this idea is much older. Coxeter, following Beecroft's idea [START_REF] Beecroft | Properties of circles in mutual contact. Lady's and Gentleman's Diary[END_REF], already noticed that the initial setting in the classic Apollonian packing with its dual can be obtained from a circumscribed octahedron [START_REF] Coxeter | The problem of Apollonius[END_REF]. One of the fundamental results which relates edge-scribed polyhedra with circle packings is the Midsphere Theorem, proved by Brightwell and Schneirman [START_REF] Brightwell | Representations of planar graphs[END_REF]. The Midsphere Theorem implies important theorems, such as the KAT and the Steinitz's theorem, and has also numerous applications in polytopes and graph drawing [START_REF] Springborn | A unique representation of polyhedral types. Centering via Möbius transformations[END_REF][START_REF] Mohar | Circle packings of maps in polynomial time[END_REF][START_REF] Felsner | On primal-dual circle representations[END_REF]. Kontorovich and Nakamura explored the integrality of the Apollonian packings based on circle packings produced by the Midsphere Theorem, which they call polyhedral packings [START_REF] Kontorovich | Geometry and arithmetic of crystallographic sphere packings[END_REF]. In higher dimensions, Eppstein, Kuperberg and Ziegler [START_REF] Eppstein | Fat 4-polytopes and fatter 3-spheres[END_REF] followed a similar approach to construct sphere packings from edge-scribed 4-polytopes which improve the upper bound of the average kissing number. Similarly, Chen studied in [START_REF] Chen | Apollonian ball packings and stacked polytopes[END_REF] the relation between Apollonian d-ball packings and stacked (d + 1)-polytopes, and explored some generalizations of the Midsphere Theorem in higher dimensions. The one-sheet hyperboloid of normalized space-like vectors S(L d+1,1 ) can be regarded in the oriented projective space of L d+1,1

P + L d+1,1 = {x ∈ L d+1,1 \ 0}/ ∼ (3.1)
where x ∼ y if there is a real λ > 0 such that x = λy. P + L d+1,1 is equivalent to the Euclidean unit sphere S d+1 ⊂ L d+1,1 which, under the gnomonic projection, becomes the union of two affine hyperplanes

Π ↑ = {x d+2 = 1} and Π ↓ = {x d+2 = -1} together with Π 0 = {(x, 0) | x ∈ S d+1 }. The composition of the isomorphism Balls( R d ) → S(L d+1,1
) with the projection

S(L d+1,1 ) → Π ↑ ∪ Π 0 ∪ Π ↓ x → x if x d+2 = 0 1 |x d+2 | x otherwise
gives an isomorphism between Balls( R d ) and Π ↑ ∪ Π 0 ∪ Π ↓ . We call the latter the projective model of Balls( R d ). We shall identify Π ↑ with the (d + 1)-dimensional Euclidean space which we shall denote by E d+1 , in order to make the difference with the compactified Euclidean space R d where the d-balls are defined. This identification induces a bijection between d-balls whose Lorentzian vector is future-directed and points of E d+1 whose Euclidean norm is strictly greater than 1. Such a point of For any d-ball whose Lorentzian vector is future-directed, the inversive coordinates can be computed from the coordinates of its light source by

i(b(u)) = 1 ∥u∥ 2 -1 u 1 (3.2)
This equality implies that for any two d-balls whose Lorentzian vectors are futuredirected, their inversive product is related to the Euclidean inner product of their corresponding light sources by the following equation

⟨b(u), b(v)⟩ = 1 (∥u∥ 2 -1)(∥v∥ 2 -1) (u • v -1) (3.3)
where • denotes the Euclidean inner product. In the projective model of space of d-balls, the Möbius group acts as the group of projective transformations preserving the unit sphere of E d+1 [106].

Polytopes

Let us quickly recall some basic polytope notions and definitions needed for the rest of the paper. We refer the reader to [START_REF] Schulte | Symmetry of polytopes and polyhedra[END_REF][START_REF] Grünbaum | Convex polytopes[END_REF] for further details. We consider here a d-polytope as the convex hull of a finite collection of points in E d . A 2-polytope and a 3-polytope are usually called polygon and polyhedron respectively.

Let d ≥ 1 and P be a (d + 1)-polytope. For every 0 ≤ k ≤ d + 1 we denote by F k (P) the set of k-faces of P and by F(P) = {∅} ∪ d+1 k=0 F k (P). A flag of P is a chain of faces (f 0 , f 1 , . . . , f d , P) where for each k = 0, . . . , d, f k ⊂ f k+1 . The elements of V (P) := F 0 (P), E(P) := F 1 (P), R(P) := F d-1 (P) and F (P) := F d (P) are called vertices, edges, ridges and facets of P, respectively. The graph of P (also called skeleton or 1-skeleton) is the graph induced by the vertices and the edges of P. The face lattice (F(P), ⊂) encodes all the combinatorial information about P. Two polytopes P and P ′ are combinatorially equivalent if there exists an isomorphism between their face lattices. If they are combinatorially equivalent, we say they have the same combinatorial type and P ′ is said to be a realization of P.

The polar of a set X ⊂ E d+1 is defined as

X * := {u ∈ E d+1 | u • v ≤ 1 for all v ∈ X}. (3.4)
If P is a (d + 1)-polytope containing the origin in its interior then P * is also a (d + 1)polytope containing the origin in its interior and holding the dual relation (P * ) * = P. There is a bijection between F(P) and F(P * ) which reverses incidences. Indeed, every facet f ∈ F d (P) corresponds to a unique vertex v f ∈ F 0 (P * ) and they are related by

f = {u ∈ P | u • v f = 1}. (3.5)
The symmetric group of P is defined as the group of Euclidean isometries of E d+1 preserving P. P is said to be regular if its symmetric group acts transitively on the set of flags P. In this case, the symmetric group of P can be generated from a flag (f 0 , . . . , f d , P) by considering the maximal simplex C (chamber) in the barycentric subdivision of P whose vertices are the barycenters of the faces in (f 0 , . . . , f d , P).

Then C is the fundamental region of the symmetric group of P which is generated by the reflections r 0 , . . . , r d , where r k is the reflection in the wall of C containing the barycenter of P and opposite to the vertex corresponding to the k-face in (f 0 , . . . , f d ).

We call the set {r 0 , . . . , r d } the fundamental generators of the symmetric group of P with respect to the flag (f 0 , . . . , f d , P). A presentation of the symmetric group of P in terms of the fundamental generators is given by

r 2 i = (r j r k ) 2 = 1 (0 ≤ i, j, k ≤ d, |j -k| ≥ 2) (r i-1 r i ) p i = 1 (0 ≤ i ≤ d)
where For every 0 ≤ k ≤ d, a (d + 1)-polytope P is said to be k-scribed if all its k-faces are tangent to the unit sphere of E d+1 . A k-scribed polytope is called inscribed, edgescribed, ridge-scribed and circumscribed if k = 0, 1, d -1, d respectively. A polytope is said to be k-scribable if it admits a realization which is k-scribed. Any regular (d + 1)polytope is k-scribable for every 0 ≤ k ≤ d (by properly centering and rescaling by a suitable value). If P is a (d + 1)-polytope containing the origin in its interior then P is k-scribed if and only if P * is (d -k)-scribed (see [START_REF] Chen | Scribability problems for polytopes[END_REF] for more general results).

For any regular (d + 1)-polytope P, we denote by ℓ P the midsphere ratio of P, obtained by the ratio between the half edge-length of P and the radius of the midsphere (the sphere tangent to every edge). The midsphere ratios of every regular polytope are presented in the Appendix (Table A.1) and were adapted from [START_REF] Coxeter | Regular Polytopes. Dover books on advanced mathematics[END_REF]. For d = 1, 2, the midsphere ratios can be computed from the Schläfli symbols by Let P be an outer-sphere (d + 1)-polytope, i.e. with all its vertices outer-sphere.

ℓ {p} = tan( π p ) ℓ {p,q} = sin 2 ( π q ) -cos 2 ( π p ) cos 2 ( π p ) (3.6)
We define the ball-arrangement projection of P, denoted by β(P), the arrangement of The d-balls in an i-CBP projection can be grouped by layers of d-balls with the same curvature. These curvatures may be expressed as a linear combination of two numbers: κ P and h i , where κ P is the mean of all the curvatures, and h i is the minimum of the positive heights, in the direction of the North Pole, among all the vertices of P. By Lemma 4.2.3, we have that κ P is equal to the inverse of the midsphere ratio of P. CBP projections are similar to the projections of regular polytopes defined by

Centered Ball Packing projections of regular polytopes

Coxeter in [START_REF] Coxeter | Regular Polytopes. Dover books on advanced mathematics[END_REF], obtained by applying parallel cross-sections. Indeed, the number of d-balls in each layer of an i-CBP projection corresponds to the number of vertices in a layer of a Coxeter's projection where the dimension of the first face in the cross-section is i.

Möbius uniqueness of edge-scribable polytopes

In Section 1.1.7, we define the notion of Möbius uniqueness for d-ball packable graphs. Here, we introduce an analogous uniqueness for edge-scribable (d + 1)polytopes. We say that an edge-scribable (d + 1)-polytope P is Möbius unique if any two polytopal d-ball packings, whose tangency polytope is an edge-scribed realization of P, are Möbius equivalent. Equivalently, P is Möbius unique if all its edge-scribed realizations are connected by a projective transformation preserving the sphere. The following algebraic approach will be useful to detect Möbius uniqueness of edge-scribable polytopes. We say that an arrangement of d-balls has maximal rank if the rank of its Gramian is equal to d + 2 (it cannot be larger). In particular, polytopal d-ball packings have maximal rank. Let V = (x 1 , . . . , x n ) and V ′ = (x ′ 1 , . . . , x ′ n ) be the Lorentzian vectors of the d-balls of B and B ′ respectively. Without loss of generality we can consider that the vectors of V and V ′ are future-directed. By the rank condition, V contains a basis

∆ = {x i 1 , . . . , x i d+2 } of L d+1,1 . Since Gram(B) = Gram(B ′ ) the corresponding col- lection ∆ ′ = {x ′ i 1 , . . . , x ′ i d+2 } ⊂ B ′
is also a basis. Let P and P ′ be the light-source polytopes of the d-balls corresponding to ∆ and ∆ ′ , respectively. Since these are basis, then the polytopes P and P ′ are outer-sphere (d + 1)-simplices intersecting the unit sphere S d ⊂ E d+1 . If P does not contain the origin, then we can find a Möbius transformation µ 0 such that the corresponding projective transformation maps P into another (d+1)-simplex P 0 containing the origin of E d+1 . Such a Möbius transformation can be easily found by combining translations and rescalings of R d which are also Möbius transformations. Let µ 0 be the element in the Orthochronous Lorentz group corresponding to µ 0 and let V 0 := µ 0 (V ), ∆ 0 := µ 0 (∆). Let M 0 and M 1 be the (d + 2)-square matrices formed by the column-matrices of the Cartesian coordinates of the vectors of ∆ 0 and ∆ ′ respectively with respect to the canonical basis of L d+1,1 .

Möbius uniqueness of edge-scribable polytopes 53

The matrices M 0 and M 1 are non-singular and therefore we have

Gram(∆ 0 ) = Gram(∆ ′ ) ⇔ M 0 T Q d+2 M 0 = M 1 T Q d+2 M 1 ⇔ Q d+2 (M 1 T ) -1 M 0 T Q d+2 M 0 = M 1
The matrix

A := Q d+2 (M 1 T ) -1 M 0 T Q d+2 defines a linear map of L d+1,1 to itself which maps ∆ 0 to ∆ ′ . Moreover, one can check that A T Q d+2 A = Q d+2
Since the origin belongs to P 0 we have that Ae d+2 ⊂ P ′ ⊂ Π ↑ . Thus,

e T d+2 Ae d+2 = A d+2,d+2 > 0
and therefore, by Equation (1.13), A ∈ O ↑ d+1,1 (R). The coordinates for the remaining vectors of V 0 and V ′ with respect to the basis ∆ 0 and ∆ ′ respectively can be obtained with the polyspherical coordinates and the entries of Gram(B). The equality of the Gramians and linearity implies that the Lorentz transformation µ given by A maps are not Möbius equivalent and therefore P is not Möbius unique.

V 0 to V ′ .
Möbius uniqueness in dimension 3 is much less restricted. Indeed, the Midsphere theorem of Brightwell and Scheinerman' [START_REF] Brightwell | Representations of planar graphs[END_REF] stating the existence and uniqueness of primal-dual circle representations implies the following.

Corollary 3.6.3 (Corollary of the Midsphere Theorem). Every polyhedron is edge-scribable and Möbius unique.

In dimension 4, there are polytopes which are not edge-scribable (see Section 3.6.2). However, we do not know if there are edge-scribable 4-polytopes which are not Möbius unique. Appart of the simplex family, we can also prove that the family of (d + 1)-cross polytopes and (d + 1)-cubes are Möbius unique for every d ≥ 2.

Before proving this, we need the following result.

⟨b u , b v ⟩ = 1 -2d G (u, v) (3.7)
where d G (u, v) is the distance between u and v in the graph of P. an antipodal labelling

V (O d+2 ) = {v 1 , . . . , v d+2 , v -1 , . . . , v -(d+2) }
where v i and v j are connected by an edge of O d+1 if and only if j ̸ = -i. For every

1 ≤ i < j ≤ d + 1,
B i,j Q d+3 X = -1 d+3 (3.9)
where B i,j is the matrix of the inversive coordinates of B i,j , Q d+3 is the matrix of the inversive product, X is a (d + 3)-column vector, and 1 d+3 is the (d + 3)-column vector of only 1's. Since b d+2 and b -(d+2) are distinct, (3.9) has more than one solution.

Therefore, B i,j is singular, which implies that there is a hyperplane H i,j of E d+2 such that

V i,j := {v 1 , . . . , v d+1 , v -i , v -j } ⊂ H i,j .
Moreover, since for every 1 ≤ j ′ ≤ d + 1, the hyperplanes H i,j and H i,j ′ share d + 2

points of E d+2 , then they must be the same hyperplane. Therefore, there is one hy-

Gram(∆) =          1 -1 • • • -1 λ -1 1 -2d . . . A d+2 . . . -1 1 -2d λ 1 -2d • • • 1 -2d 1         
where A n denotes the n-square matrix with 1's in the diagonal entries and -3 everywhere else and λ := ⟨b u , b ū⟩. It can be computed that

det(Gram(∆)) = -4 d+1 (d(2d -3λ + 3) -2λ + 2)(2d + λ + 3).
Since ∆ corresponds to a collection of d + 4 vectors of L d+2,1 the Gramian of ∆ must be singular. Therefore, by the above equality, we obtain

det(Gram(∆)) = 0 ⇔ λ = -3 -2d or λ = 2d 2 3d + 2 + 1 > 0 for every d ≥ 2.

The Möbius spectra

Spectral techniques, based on the eigenvalues and the eigenvectors of the adjacency or the Laplace matrices of graphs are strong and successful tools to investigate different graph's properties. In this spirit, we define the Möbius spectra of an edgescribable Möbius unique (d + 1)-polytope P as the multiset of the eigenvalues of the Gramian of β(P ′ ) where P ′ is any edge-scribed realization of P. We believe that Möbius spectra may shed light on the properties of edge-scribable polytopes. In the appendices, we show in Table A.1 the Möbius spectra of the regular polytopes which we know are Möbius unique.

Non edge-scribable 4-polytopes.

The first examples of 4-polytopes not admitting an edge-scribed realization were given by Schulte in [START_REF] Schulte | Analogues of Steinitz's theorem about non-inscribable polytopes[END_REF]. In [START_REF] Eppstein | Fat 4-polytopes and fatter 3-spheres[END_REF]Corollary 9], Eppstein, Kupperberg and Ziegler found another type of non edge-scribable 4-polytope by noticing that every stacked 4-polytope with more than 6 vertices is not edge-scribable. We recall that the connected sum of two polytopes P and P ′ along a facet f , denoted by P# f P ′ , is the glueing of P and P ′ along f where one of them is projectively deformed in such a way that the faces of P# f P ′ are the union of the faces of P and P ′ minus f . A In this packing, the dual sphere b f 0 , corresponding to the facet f 0 of T 1 with vertices {1, 2, 3, 4}, cuts orthogonally the sphere b 7 . By combining Equations (3.5), (3.3) and (1.5), we have that the vertex v 7 of T 3 lies in the affine hull f 0 , so f 0 is not a face of (T # f 1 T ′ )# f 2 T ′′ . This contradicts the condition on the set of faces in the definition of connected sum. Therefore, • The connected sum of two orthoplexes.

(T # f 1 T ′ )# f 2 T ′′ is not edge-scribable.
• The connected sum of three hypercubes sharing a ridge.

• The connected sum of three 24-cell's sharing a ridge.

Proof. We may apply the same arguments of the proof of Lemma 3.6.1. By the Möbius uniqueness of the orthoplex, hypercube and 24-cell and Lemma 1.1.1, after applying the corresponding inversion and a proper rescaling, we must obtain a packing which is Euclidean similar to the glueing by reflections of 1-CBP projections of the corresponding 4-polytope. Then, as above, we would have a dual sphere cutting orthogonally spheres of the different components in the connected sum, and the same contradiction arises.

Apollonian groups and packings

In this section, we introduce a generalization of the Apollonian group introduced by Graham et al. in [START_REF] Graham | Apollonian circle packings: Geometry and group theory i. the Apollonian group[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory II. super-Apollonian group and integral packings[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory III. higher dimensions[END_REF] and Lagarias et al. in [START_REF] Lagarias | Beyond the Descartes circle theorem[END_REF] for any polytopal d-ball packing.

For d ≥ 2, let B P a polytopal d-ball packing. We consider the following subgroups of Möb( R d ):

• The symmetric group of B P is the group

Sym(B P ) := ⟨µ ∈ Möb( R d ) | µ(B P ) = B P ⟩.
We notice that Sym(B P ) = Sym(B * P ).

• The Apollonian group of B P is the group

A(B P ) := ⟨S(B * P )⟩
where S(B * P ) denotes the set of inversions on the d-balls of B * P .

• The symmetrized Apollonian group of B P is the group

SA(B P ) := ⟨Sym(B P ) ∪ A(B P )⟩
It can be checked that for every r ∈ Sym(B P ) and every b ∈ B * P we have

rs b r = s r(b) ∈ A(B P ) (3.10)
where s b denotes the inversion of b. And consequently, we have that SA(B P ) =

A(B P ) ⋊ Sym(B P ).

• The super symmetrized Apollonian group of B P is the group SSA(B P ) := ⟨A(B * P ) ∪ Sym(B P ) ∪ A(B P )⟩.

Apollonian groups and packings 61

By definition, for any polytopal d-ball packing A(B P ) < SA(B P ) < SSA(B P ).

Therefore, we can find representations of A(B P ) as a subgroup of SSA(B P ). Clearly, SSA(B P ) might have a smaller set of generators than A(B P ). In [START_REF] Graham | Apollonian circle packings: Geometry and group theory i. the Apollonian group[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory II. super-Apollonian group and integral packings[END_REF][START_REF] Graham | Apollonian circle packings: Geometry and group theory III. higher dimensions[END_REF][START_REF] Kontorovich | Geometry and arithmetic of crystallographic sphere packings[END_REF] the authors defined the dual Apollonian group and the super Apollonian group which, with our notation, correspond to the groups A(B * P ) and ⟨A(B P ) ∪ A(B * P )⟩, respectively. For the purposes of this manuscript, these groups are not needed. Nevertheless, both groups are subgroups of SSA(B P ).

For any other polytopal d-ball packing B ′ P Möbius equivalent to B P the four groups defined above for B P are congruent to their analogues in B ′ P . Therefore, if P is Möbius unique, the four groups can be defined for P independently of the choice of B P , up to isomorphism.

Since inversions of Möb( R d ) can be seen as hyperbolic reflections of the (d + 1)dimensional hyperbolic space, Apollonian groups are hyperbolic Coxeter groups. This point of view has been well-explored in [START_REF] Chen | Even more infinite ball packings from Lorentzian root systems[END_REF][START_REF] Chen | Lorentzian Coxeter systems and Boyd-Maxwell ball packings[END_REF]. The Apollonian cluster of B P , denoted by Ω(B P ), is the union of the orbits of the action of A(B P ) on B P . If Ω(B P ) is a packing we shall call it the Apollonian packing of B P . For d = 2, Apollonian clusters are always packings. However, this is not true in higher dimensions. For instance, for d ≥ 4, the Apollonian cluster Ω(B P ) where P is an edge-scribed (d + 1)-simplex, is not a packing [START_REF] Graham | Apollonian circle packings: Geometry and group theory III. higher dimensions[END_REF].

The four groups presented above give different actions on the Apollonian cluster. Indeed, each disk of a polytopal circle packing B P has a different orbit under the action of A(B P ). This can be used to present colorings of the disks in the Apollonian packing of B P . Indeed, if B P admits a proper coloring, then is not hard to see that the coloring can be extended to Ω(B P ) by the action of the A(B P ). We present in Figure

The Platonic Apollonian Groups

In this section, we compute the Apollonian groups of the Platonic solids, which we shall call the Platonic Apollonian groups. Theorem 3.7.1. Let B P be a polytopal circle packing where P is a Platonic solid with Schläfli symbol {p, q}. The group SSA(B P ) is the hyperbolic Coxeter group with Coxeter graph ∞ p q ∞ . Moreover, SSA(B P ) is generated by the following five matrices:

S =       -1 1 1 1       , V =       1 -1 1 1       , E =       1 0 0 0 0 1 /2 1 -1 /2 0 1 -1 1 0 1 /2 -1 3 /2       F q =       -1 0 4 cos( π q ) -4 cos( π q ) 0 1 0 0 4 cos( π q ) 0 1 -8 cos 2 ( π q ) 8 cos 2 ( π q ) 4 cos( π q ) 0 -8 cos 2 ( π q ) 1 + 8 cos 2 ( π q )       , S * =       1 0 0 0 0 -1 -2 2 0 -2 -1 2 0 -2 -2 3      
Proof. We follow a similar strategy as in the proof of Theorem 3.7.2. For every q = 3, 4, 5, let [B P ] 1 2 be the standard polytopal circle packing where P is the regular polyhedron of Schläfli symbol {3, q}. We choose the flag (v, e, f ) satisfying the same conditions as in the proof of Theorem 3.7.2. Let us now apply a translation and a reflection if needed so b f becomes the half-space {x ≥ 0}. Let B 3,q be the circle packing obtained after the transformations. Since P is regular, SSA(B 3,q ) is generated by {s v , r v , r e , r f , s f } where {r v , r e , r f } are the elements in Sym(B 3,q ) corresponding to the fundamental generators of the symmetric group of P with respect to the flag (v, e, f ), s f is the inversion on the disk in B * 3,q corresponding to f and s v is the inversion on the disk in B 3,q corresponding to v. For each q = 3, 4, 5 we have that • s v is the reflection on the line {y = 1},

• r v is the reflection on the line {y = 0},

• r e is the inversion on the circle centered at (1, 1) and radius 2,

• r f is the reflection on the line {x = -2 cos( π q )},

• s f is the reflection on the line {x = 0}.

s v s f r v r f r e s v s f r v r f r e s v s f r v r f r e Figure 3
.8: The circle packings B 3,q with their dual (in blue) for q = 3 (left), q = 4 (center) and q = 5 (right). In dashed line the generators of the symmetric group of B 3,q . The central disk is a unit disk centered at the origin.

By using the inversive coordinates and Eq. (1.14) we obtain a faithful linear representation of SSA(B 3,q ) as a discrete subgroup of O ↑ 3,1 (R) where the generators are mapped to:

s v → S * r v → V r e → E r f → F q s f → S
The relations of finite order in the Coxeter graph can be checked by straightforward computations on the matrices. For every n ∈ N the last entry in the diagonal of (S * V) n and (F q S) n is, respectively, 1 + 2n 2 and 1 + 8 cos 2 ( π q )n 2 and therefore every product has infinite order for every q = 3, 4, 5. Since SSA(B 3,q ) = SSA(B * 3,q ) then the results also hold for the cube and the dodecahedron. Indeed, the generators of SSA(B * 3,q ) are given by swapping r v with r f and s v with s f .

The previous theorem allows to exhibit a matrix representation of the Platonic Apollonian groups in O ↑ 3,1 (K) where K is one of these 3 number fields: Z, Z[

√ 2] and Z[φ].
We illustrates this for the tetrahedron, octahedron and cube.

Corollary 3.7.1. The tetrahedral Apollonian group is isomorphic to

⟨T 1 , T 2 , T 3 , T 4 ⟩ < O ↑ 3,1 (Z)
where

T 4 =       -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1       = S T 3 =       -1 0 4 -4 0 1 0 0 4 0 -7 8 4 0 -8 9       = F 3 T 4 F 3 T 2 =       -1 2 0 -2 2 -1 0 2 0 0 1 0 2 -2 0 3       = ET 3 E T 1 =       -1 -2 0 -2 -2 -1 0 -2 0 0 1 0 2 2 0 3       = VT 2 V
C 123 =     -1 1 1 1     = S C 123 =     -1 0 4 √ 2 -4 √ 2 0 1 0 0 4 √ 2 0 -15 16 4 √ 2 0 -16 17     = F 4 C 123 F 4 C 123 =     -1 2 √ 2 0 -2 √ 2 2 √ 2 -3 0 4 0 0 1 0 2 √ 2 -4 0 5     = EC 123 E C 123 =     -1 -2 √ 2 0 -2 √ 2 -2 √ 2 -3 0 -4 0 0 1 0 2 √ 2 4 0 5     = VC 123 V C 123 =     -17 6 √ 2 12 √ 2 -18 √ 2 6 √ 2 -3 -8 12
    = VC 123 V C 123 =     -17 0 0 -12 √ 2 0 1 0 0 0 0 1 0 12 √ 2 0 0 17     = EC 123 E C 123 =     -49 0 20 √ 2 -40 √ 2 0 1 0 0 20 √ 2 0 -15 32 40 √ 2 0 -32 65     = F 4 C 123 F 4
Corollary 3.7.3. The cubical Apollonian group is isomorphic to

⟨O 1 , O 2 , O 3 , O 3 , O 2 , O 1 ⟩ < O ↑ 3,1 (Z[ √ 2])
where

O 1 =     1 0 0 0 0 -1 -2 2 0 -2 -1 2 0 -2 -2 3     = S * O 2 =     1 0 0 0 0 -1 2 -2 0 2 -1 2 0 2 -2 3     = VO 1 V O 3 =     1 1 -1 1     = EO 2 E O 3 =     -15 0 12 √ 2 -16 √ 2 0 1 0 0 12 √ 2 0 -17 24
16 √ 2 0 -24 33     = F 4 O 3 F 4 O 2 =     -15 4 √ 2 4 √ 2 -12 √ 2 4 √ 2 -1 -2 6 4 √ 2 -2 -1 6 12 √ 2 -6 -6 19     = EO 3 E O 1 =     -15 -4 √ 2 4 √ 2 -12 √ 2 -4 √ 2 -1 2 -6 4 √ 2 2 -1 6 12 √ 2 6 -6 19     = VO 2 V

Chapter 3. Polytopal ball packings

We end this chapter with the following result.

Theorem 3.7.2. There is a tetrahedral, cubical and dodecahedral Apollonian packing where the set of curvatures of the disks contains all the perfect squares.

Proof. For every p = 3, 4, 5, let [B P ] 1 2 be the standard polytopal circle packing, where P is the regular polyhedron of Schläfli symbol {p, 3}. Let (v, e, f ) be the flag of P where v is the vertex corresponding to the half-space b v = {y ≤ -1}, e is the edge of P with ends v and v ′ where b v ′ = {y ≤ -1}, and f is a face of P containing e.

Let us apply a translation and a reflection if needed, so b f becomes the half-space {x ≥ 0}. We then rescale by a factor of λ = 4 cos 2 ( π p ). Let B {p,3} , for p = 3, 4, 5, be the transformed circle packing obtained from [B P ] 1 2 (Fig. 3.9). The rescaling factor has been chosen to obtain that the disks with minimal non-zero curvature of B {p,3} have curvature equal to one. Since P is regular then SA(B {p,3} ) is generated by {r v , r e , r f , s f } where {r v , r e , r f } are the elements in Sym(B {p,3} ) corresponding to the fundamental generators of the symmetric group of P with respect to the flag (v, e, f ) and s f is the inversion on the b f . Therefore, for p = 3, 4, 5, we have the following:

r f r v b v ′ b v s f r e r f r v b v ′ b v s f r e r f r v b v ′ b v s f r e
• b v is the half-space {y ≤ -λ} and b v ′ is the half-space {y ≥ λ},

• r v is the reflection on the line {y = 0},

Apollonian groups and packings

67

• r e is the inversion on the circle centered at (0, -λ) and radius 2 √ λ,

• r f is the reflection on the line {x = -√ λ},

• s f is the reflection on the line {x = 0}.

For every p = 3, 4, 5 and for every integer n ≥ 0 we define the element

µ p,n := r e (r f s f ) n r e ∈ SA(B {p,3} ) and the disk b n := µ p,n (b v ′ ) ∈ Ω(B {p,3} ).
Let us compute the curvature of b n . The inversive coordinates of b v ′ and the matrices representing r e , r f and s f obtained by Eq. (1.14), are given by

i(b v ′ ) =       0 1 λ λ       , r e → E λ =        1 0 0 0 0 1 -λ 2 1 4 λ 2 -4λ -1 -1 4 λ 2 -4λ + 1 0 1 4 λ 2 -4λ -1 1 - (λ 2 -4λ-1) 2 8λ -1-(λ-4) 2 λ 2 8λ 0 1 4 λ 2 -4λ + 1 1-(λ-4) 2 λ 2 8λ 1 + (λ 2 -4λ+1) 2 8λ        , r f → F λ =       -1 0 2 √ λ -2 √ λ 0 1 0 0 2 √ λ 0 1 -2λ 2λ 2 √ λ 0 -2λ 2λ + 1       , s f → S =       -1 1 1 1      
, By induction on n we obtain

(F λ S) n =       1 0 2 √ λn -2 √ λn 0 1 0 0 -2 √ λn 0 1 -2λn 2 2λn 2 -2 √ λn 0 -2λn 2 1 + 2λn 2      
and therefore, by Eq. (1.9), we have

κ(b n ) = κ(µ p,n (b v ′ )) = k 4 E λ (F λ S) n E λ i(b v ′ ) = n 2
where k 4 = 0 0 -1 1 .

Introduction

The classical Apollonian circle packing is constructed from an initial configuration of four pairwise tangent disks in R 2 , also known as a Descartes configuration [START_REF] Chen | Apollonian ball packings and stacked polytopes[END_REF]. The name comes from René Descartes who stated, in his correspondence with the Princess Elizabeth of Bohemia in 1643, an algebraic relation equivalent to the following theorem [START_REF] Bos | Princess Elizabeth of Bohemia and Descartes' letters (1650-1665)[END_REF].

Theorem 4.1.1 (Descartes). The curvatures of four pairwise tangent disks in the plane satisfy

(κ 1 + κ 2 + κ 3 + κ 4 ) 2 = 2(κ 2 1 + κ 2 2 + κ 2 3 + κ 2 4 ) (4.1)
In 1936, Sir Frederick Soddy 1 published in Nature a paper extending the Descartes' Theorem in dimension three. One year later, Thorold Gosset presented in [START_REF] Gosset | The Hexlet[END_REF] a proof for a generalization of the Descartes' Theorem in arbitrary dimensions.

(κ 1 + • • • + κ d+2 ) 2 = d κ 2 1 + • • • + κ 2 d+2 (4.2)
Soddy-Gosset's theorem was discovered and rediscovered many times [START_REF] Lagarias | Beyond the Descartes circle theorem[END_REF][START_REF] Lachlan | on systems of circles and spheres[END_REF][START_REF] Beecroft | Properties of circles in mutual contact. Lady's and Gentleman's Diary[END_REF].

A different type of generalization of the Descartes' Theorem was given by Guettler and Mallows in [START_REF] Guettler | A generalization of Apollonian packing of circles[END_REF] for octahedral circle packings.

Theorem 4.1.3 (Guettler-Mallows). Let κ 1 , κ 2 , κ 3 , κ -1 , κ -2 , κ -3 be the curvatures of the disks in an octahedral circle packing where the labelling satisfies that κ i and κ -i are the curvatures of disjoint disks. Then, the curvatures satisfy the following relations

κ 1 + κ -1 = κ 2 + κ -2 = κ 3 + κ -3 =: 2κ µ (4.3) κ 2 µ -2(κ 1 + κ 2 + κ 3 )κ µ + (κ 2 1 + κ 2 2 + κ 2 3 ) = 0 (4.4)
In two independant works, Dias and Nakamura push forward the Guettler and Mallows' generalization to orthoplicial sphere packings, the analogue of octahedral circle packings in dimension 3. We shall study in detail these packings in Chapter 5.

Theorem 4.1.4 (Dias-Nakamura). Let κ 1 , κ 2 , κ 3 , κ 4 κ -1 , κ -2 , κ -3 , κ -4 be the curvatures of the spheres in an orthoplicial sphere packing where the labelling satisfies that κ i and κ -i are the curvatures of disjoint spheres. Then, the curvatures satisfy the following relations

κ 1 + κ -1 = κ 2 + κ -2 = κ 3 + κ -3 = κ 4 + κ -4 =: 2κ µ (4.5) κ 2 µ -2(κ 1 + κ 2 + κ 3 + κ 4 )κ µ + (κ 2 1 + κ 2 2 + κ 2 3 + κ 2 4 ) = 0 (4.6)
The Descartes' Theorem and its generalizations are a key ingredient for the construction of integral Apollonian packings. In this chapter, we present a generalization of the Descartes' Theorem for all the polytopal d-ball packings whose tangency polytope is regular. We call this new generalization the Polytopal Descartes' Theorem. We will see that the Descartes' Theorem can be obtained as a particular case of

The Polytopal Descartes' Theorem

In the background given in Chapter 1, we explained that the curvature of a dball b can be obtained from the Lorentzian product of the corresponding space-like normalized vector x b ∈ S(L d+1,1 ) with a specific vector x N = e d+1 + e d+2 where e i denotes the i-th vector of the canonical basis B 0 of L d+1,1 (Eq. (1.9)). Let us extend this notion of curvature to any vector x ∈ L d+1,1 by

κ(x) = -⟨x N , x⟩ (4.7)
Let P ⊂ E d+1 be an outer-sphere polytope. For every face f of P, we define the Lorentzian barycenter of f as the vector x f ∈ L d+1,1 given by

x f := 1 |V (f )| v∈V (f ) x b(v) (4.8)
where V (f ) is the set of vertices of f and b(v) is the illuminated region of v. We also define the Lorentzian curvature of f as κ f := κ(x f ). By linearity, we have that

κ f = 1 |V (f )| v∈V (f ) κ(b(v)) (4.9)
Let us now suppose that P is edge-scribed and B P is a polytopal d-ball packing.

For every f of P, we shall say that κ f is a polytopal curvature of B P . The polytopal curvature of a vertex v is exactly the curvature of the corresponding d-ball b v of B P .

A first relation between the polytopal curvatures of polytopal d-ball packing is the following.

Lemma 4.2.1 (Antipodal relation).

Let B P be a polytopal d-ball packing where P is a regular edge-scribed (d + 1)-polytope which is centrally symmetric. Then, for any two vertices v, v at maximal distance in the 1-skeleton of P, we have

κ P = κ v + κ v 2
Proof. Since P is centrally symmetric, then 1 2 (x bv + x bv ) is the Lorentzian barycenter of P. The Lemma follows from Eq. (4.7) and linearity.

The above lemma applies to every centrally symmetric regular (d + 1)-polytope.

These are the p-gons with p even when d = 1 and all the regular polytopes not belonging to the simplex family when d ≥ 2.

We define the flag quadratic form of a regular (d + 1)-polytope P as

Φ P (x 0 , x 1 , . . . , x d+1 ) := d i=0 (x i+1 -x i ) 2 L P (i + 1) -L P (i) - x 2 d+1 L P (d + 1) (4.10)
where

L P (i) :=        -1 i = 0 0 i = 1 ℓ -2 f i if 2 ≤ i ≤ d + 1
and ℓ f i denotes the midsphere ratio of f i . From the values of the midsphere ratio given in Table A.1, one can compute that, for every d ≥ 1, the flag quadratic form of the (d + 1)-simplex, (d + 1)-cross polytope and (d + 1)-cube is

Φ T d+1 (x 0 , . . . , x d+1 ) = d i=0 i + 2 2 (x i -x i+1 ) 2 - d + 2 d x 2 d+1 (4.11) Φ O d+1 (x 0 , . . . , x d+1 ) = d-1 i=0 i + 2 2 (x i -x i+1 ) 2 + d + 1 2 (x d -x d+1 ) 2 -x 2 d+1 (4.12) Φ C d+1 (x 0 , . . . , x d+1 ) = d i=0 (x i -x i+1 ) 2 - 1 d x 2 d+1 . (4.13)
We now can state the main theorem. The following lemmas are needed for the proof of Theorem 4.2.1.

Lemma 4.2.2. Let ∆ = (x 1 , . . . , x d+2 ) be a collection of d + 2 vectors in L d+1,1 . Then ∆ is a basis if and only if Gram(∆) is non-singular. Moreover, the vector (κ 1 , . . . , κ d+2 ) where

κ i := κ(x i ) satisfies (κ 1 , . . . , κ d+2 ) Gram(∆) -1     κ 1 . . . κ d+2     = 0 (4.15)
Proof. Let M be the matrix whose columns are the coordinates of the vectors of ∆ with respect to the canonical basis of L d+1,1 , N = (0, . . . , 0, 1, 1) T and Q d+2 = Diag(1, . . . , 1, -1). We have that (κ 1 , . . . , κ d+2 ) = (κ(x 1 ), . . . , κ(

x d+2 )) = -M T Q d+2 N.
Therefore,

(κ 1 , . . . , κ d+2 ) Gram(∆) -1     κ 1 . . . κ d+2     = (-N T Q d+2 M)(M -1 Q d+2 (M T ) -1 )(-M T Q d+2 N) = N T Q d+2 N = 0
As Boyd noticed in [START_REF] Boyd | A new class of infinite sphere packings[END_REF], the Soddy-Gosset's generalization (Th. 4.1.2) follows directly by applying the above lemma to the Lorentzian vectors of a packing of d + 2

pairwise tangent d-balls.

Lemma 4.2.3. Let P be a regular edge-scribed (d + 1)-polytope with d ≥ 1. The Lorentzian vectors of the vertices of P are contained in the hyperplane {x d+2 = 1 ℓ P } where ℓ P is the midsphere ratio of P. Furthermore, for every f ∈ F(P) we have ⟨x f -x P , x P ⟩ = 0.

Proof. Since P is regular and edge-scribed the barycenter of P is at the origin of E d+1 and any vertex v of P has the same Euclidean norm ∥v∥. Therefore, the length of each half edge of P is equal to ℓ P = ∥v∥ 2 -1. By Eq. (3.2), the Lorentzian vectors of the vertices of P are contained in the hyperplane

H = {x d+2 = 1 √ ∥v∥ 2 -1 = 1 ℓ P } ⊂ L d+1,1
implying that for every face f of P the Lorentzian vector x f is also in H. Since the barycenter of P is the origin of E d+1 then x P = 1 ℓ P e d+2 implying the second part of the Lemma.

Chapter 4. The Polytopal Descartes' Theorem

Let a 1 , . . . , a n be a sequence of real numbers. We define the corner matrix C(a 1 , . . . , a n ) as the matrix with the following shape

C(a 1 , . . . , a n ) :=             a 1 a 2 a 3 • • • a n-1 a n a 2 a 2 a 3 • • • a n-1 a n a 3 a 3 a 3 • • • a n-1 a n . . . . . . . . . . . . . . . . . . a n-1 a n-1 a n-1 • • • a n-1 a n a n a n a n • • • a n a n             (4.16)
If a i ̸ = a i+1 for each i = 1, . . . , n-1 then C(a 1 , . . . , a n ) is non-singular. It can be easily checked that its inverse is equal to

                    1 a 1 -a 2 -1 a 1 -a 2 0 • • • 0 0 0 -1 a 1 -a 2 1 a 1 -a 2 + 1 a 2 -a 3 -1 a 2 -a 3 . . . 0 0 0 0 -1 a 2 -a 3 1 a 2 -a 3 + 1 a 3 -a 4 . . . . . . 0
1 a n-3 -a n-2 + 1 a n-2 -a n-1 -1 a n-2 -a n-1 0 0 0 0 . . . -1 a n-2 -a n-1 1 a n-2 -a n-1 + 1 a n-1 -an -1 a n-1 -an 0 0 0 • • • 0 -1 a n-1 -an 1 a n-1 -an + 1 an                    
Moreover, if X n = (x 1 , . . . , x n ) then it can also be checked that The desired equality will then follow from this as explained below. Since Gramians and corner matrices are symmetric, then Eq. (4.18) is equivalent to for any d ≥ 1, Gram(∆) i,j = -L P (j -1) for every

X n C(a 1 , . . . , a n ) -1 X T n = n-1 i=1 (x i -x i+1 ) 2 a i -a i+1 + x 2 n a n (4.17) Proof of Theorem 4.2.1. Let (f 0 = v, f 1 = e, . . . , f d = f, f d+1 = P)
1 ≤ i ≤ j ≤ d + 2 (4.19)
where Gram(∆) i,j denotes the (i, j)-entry of Gram(∆). We shall consider three cases according to the value of j in (4.19):

(i) j = 1, 2. Let v ′ be the other vertex of e. Then,

• Gram(∆) 1,1 = ⟨x v , x v ⟩ = ⟨x b(v) , x b(v) ⟩ = 1 = -L P (0). • Gram(∆) 1,2 = ⟨x v , x e ⟩ = ⟨x v , 1 2 (x v + x v ′ )⟩ = 1 2 (1 -1) = 0 = -L P (1). • Gram(∆) 2,2 = ⟨x e , x e ⟩ = ⟨ 1 2 (x v + x v ′ ), 1 2 (x v + x v ′ )⟩ = 1 4 (1 -2 + 1) = 0 = -L P (1).
(ii) j = d + 2. Since P is a regular edge-scribed realization, the first part of the Lemma 4.2.3 implies that x P = 1 ℓ P e d+2 where e d+2 is the (d + 2)-th vector of the canonical basis of L d+1,1 . Therefore,

• Gram(∆) d+2,d+2 = ⟨x P , x P ⟩ = -ℓ -2 P = -L P (d + 1).

The second part of Lemma 4.2.3 implies that for each i = 1, . . . , d + 2 we have

• Gram(∆) i,d+2 = ⟨x f i-1 , x P ⟩ = ⟨x P , x P ⟩ = -L P (d + 1). (iii) 3 ≤ j ≤ d + 2. If d =
⟨x f i-1 , x f d ⟩ = -ℓ -2 f d for every 1 ≤ i ≤ d + 1
where f i is the i-polytope obtained from µ(f i ) after the identification {x d+1 = 0} ≃ E d . This identification preserves the inversive product of the illuminated regions of the outer-sphere points lying in {x d+1 = 0}. Since the inversive product is also invariant under Möbius transformations then, for each i = 1, . . . , d + 1, we have:

• Gram(∆) i,d+1 = ⟨x f i-1 , x f d ⟩ = ⟨x f i-1 , x f d ⟩ = -L P (d).
Then, to prove (4.19) for j = d we apply the same arguments as above by consid- Finally, since the sequence

(-L P (0), -L P (1), . . . , -L P (d + 1)) = (1, 0, -ℓ -2 f 2 , . . . , -ℓ -2 P )
is strictly decreasing then Gram(∆) is non-singular. Therefore, by applying the Lemma 4.2.2 to ∆ we obtain

0 = (κ f 0 , . . . , κ f d+1 ) Gram(∆) -1 (κ f 0 , . . . , κ f d+1 ) T = (κ f 0 , . . . , κ f d+1 )C(-L P (0), . . . , -L P (d + 1)) -1 (κ f 0 , . . . , κ f d+1 ) T = d i=0 (κ f i+1 -κ f i ) 2 L P (i + 1) -L P (i) - κ 2 P L P (d + 1) = Φ P (κ v ,
κ P ± = ℓ f ℓ P 2 κ f ± ℓ -2 P ℓ 2 f -ℓ 2 P Φ f (κ v , . . . , κ f ) (4.20)
Proof. It follows from the definition (4.10) that

Φ P (x 0 , x 1 , . . . , x d+1 ) = Φ f (x 0 , x 1 , . . . , x d ) - ℓ 2 f x d -ℓ 2 P x d+1 2 ℓ 2 f -ℓ 2 P (4.21)
By combining this with (4.14) and then resolving for κ P we obtain (4.20).

Let us explain in detail why Theorem 4.2.1 generalizes the Descartes' Theorem.

We define the simplicial, hyperoctahedral and hypercubical quadratic form as

T d+1 (u 1 , . . . , u d+2 ) :=Φ T d+1 u 1 , 1 2 (u 1 + u 2 ), . . . , 1 d + 2 (u 1 + . . . + u d+2 ) (4.22) = 1 2 1 d ( d+2 i=1 u i ) 2 - d+2 i=1 u 2 i O d+1 (u 1 , . . . , u d+2 ) :=Φ O d+1 u 1 , 1 2 (u 1 + u 2 ), . . . , 1 d + 1 (u 1 + . . . + u d+1 ), u d+2 (4.23) =u 2 d+2 - 1 2 d+1 i=1 (u i -u d+2 ) 2 C d+1 (u 1 , . . . , u d+2 ) :=Φ C d+1 u 1 , 1 2 (u 1 + u 2 ), . . . , 1 2 (u 1 + u d+2 ) (4.24) = 1 4 1 d (u 1 + u d+2 ) 2 - d+1 i=1 (u i -u i+1 ) 2
d d+2 i=1 κ 2 i = ( d+2 i=1 κ i ) 2 (4.25)
Proof. Since T d+1 is Möbius unique, the Polytopal Descartes' Theorem holds for any polytopal d-ball packing B T d+1 . For every i = 0, . . . , d + 1, let v i be the vertex of

T d+1 corresponding to the d-ball of curvature κ i . Since T d+1 is a (d + 1)-neighborly
polytope, every set of vertices span a face. Thus, we can find a flag (f 0 , . . . , f d , f d+1 = T d+1 ) where the vertices of f i are v 0 , . . . , v i . Therefore, we have

κ f i = 1 i (κ 1 + . . . + κ i ). (4.26)
Then, by the Polytopal Descartes' Theorem,

T d+1 (κ 1 , . . . , κ d+2 ) = Φ T d+1 (κ f 0 , . . . , κ f d+1 ) = 0
which is equivalent to (4.25).

Chapter 4. The Polytopal Descartes' Theorem

In the above corollary, "polytopal d-ball packing B T d+1 " can be replaced by a "d-ball packing made by d+2 pairwise tangent d-balls", as in the original statement of Th.

(κ i -κ O d+1 ) 2 = 2κ 2 O d+1 (4.27)
Proof. We can apply the same arguments as in the proof of Corollary 4.2.

(κ i -κ i+1 ) 2 = (κ 1 + κ d+2 ) 2 . (4.28)
Proof. By Möbius uniqueness, we can consider that C d+1 is regular. Let (v 1 , . . . , v d+2 ) be a path passing through the vertices of C d+1 corresponding to the curvatures.

There is a flag (f 0 , . . . , f d+1 ) where f i is the unique i-face containing the vertices v 1 , . . . , v i . For every i = 2, . . . , d + 1, the intersection f i ∩ S d gives a polytopal i-ball packing B C i . By applying the Antipodal relation to B C i+1 we obtain that, for every i = 1, . . . , d + 2,

κ f i = 1 2 (κ 1 + κ i ). (4.29)

The Polytopal Descartes' Theorem 79

and therefore,

C d+1 (κ 1 , . . . , κ d+2 ) = Φ C d+1 (κ f 0 , . . . , κ f d , κ f d+1 ) = 0
which is equivalent to (4.28).

In the case when all the curvatures in the d-ball packing in Corollaries 4. -Relation between two consecutive vertices:

d(m 2 1 + • • • + m 2 d+2 ) = n 2 (4.30) m 2 1 + • • • + m 2 d+1 = 2n 2 (4.31) d(m 2 1 + • • • + m 2 d+1 ) = n 2 (4.
κ v + κ v ′ 2 = κ e (4.34)
where v ′ is the vertex in V (P) \ {v} satisfying v ′ ⊂ e.

-Relation between two consecutive edges:

κ e + κ e ′ 2 = cos( π p ) 2 κ v + sin( π p ) 2 κ f (4.35)
where e ′ is the edge in E(P) \ {e} satisfying v ⊂ e ′ ⊂ f .

-Relation between two consecutive faces:

κ f + κ f ′ 2 = cos 2 ( π q ) sin 2 ( π p ) κ e + sin 2 ( π q ) -cos 2 ( π p ) sin 2 ( π p ) κ P (4.36)
where f ′ is the face in F (P) \ {f } satisfying e ⊂ f ′ ⊂ P.

-Relation between two consecutive polyhedra:

κ P + κ P ′ 2 = sin 2 ( π p ) sin 2 ( π p ) -cos 2 ( π q ) κ f (4.37)
where κ P ′ is the polytopal curvature of B P ′ := s f (B P ) and s f is the inversion on the disk b f ∈ B * P .

Proof. Each relation follows by first combining Eq. (3.6) with the Polytopal Descartes' Theorem, then resolving explicitly the quadratic equation for κ v , κ e , κ f and κ P , respectively, and finally taking the arithmetic mean of both solutions. Proof. Let κ i-1 , κ i and κ i+1 be the curvatures of three consecutive tangent disks d i-1 , d i , d i+1 ∈ B P corresponding to three vertices of P lying in the same face f . Let e and e ′ be the edges of P with vertices v i-1 , v i and v i , v i+1 , respectively. By replacing

κ e = 1 2 (κ i + κ i+1 )
and κ e ′ = 1 2 (κ i + κ i-1 ) in Eq. (4.35), we obtain

κ f = 1 4 sin 2 ( π p ) (κ i+1 + κ i-1 ) + (1 - 1 2 sin 2 ( π p ) )κ i (4.38)
By replacing κ v by κ i , κ e by 1 2 (κ i + κ i+1 ) and κ f by the right-hand side of (4.38) in Eq. (4.20), and then combining with Eq. (3.6), we obtain

κ P ± = -cos( 2π p )κ i + 1 2 (κ i+1 + κ i-1 ) ± cos( π q ) (1 -4 cos 2 ( π p ))κ 2 i + κ i κ i+1 + κ i κ i-1 + κ i+1 κ i-1 2(sin 2 ( π q ) -cos 2 ( π p )) (4.39)
where the two values correspond to the polytopal curvature κ P and κ P ′ of B P and s f (B P ), respectively. If we fix κ P , we can obtain the curvatures of the disks corresponding to the vertices of P by iterating Eq. (4.39). If, instead, we fix κ(P ′ ), we obtain the curvatures of the disks corresponding to the vertices of P ′ . By changing faces and polyhedra we can obtain the curvatures for all the disks in Ω(B P ) in terms of κ i , κ i+1 and κ i-1 .

Octahedral Apollonian packings

In the same paper where Guettler and Mallows presented the octahedral Descartes' Theorem, they were able to construct integral octahedral Apollonian packings by giving an integrality condition. Here, we review their result and we prove the reciprocal, which will be useful in the next chapter. Then

Ω(B O 3 ) is integral if and only if κ 1 , κ 2 , κ 3 and 2T 2 (κ 1 , κ 2 , κ 3 ) are integers.
Proof. We mimic the same method used by Nakamura for orthoplicial Apollonian packings (see [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF]). Let us first suppose that κ 1 , κ 2 , κ 3 and 2T 2 (κ 1 , κ 2 , κ 3 ) are integers. Then, by Cor. 4.2.1, we have that κ O 3 is equal to one of the following two expressions

κ 1 + κ 2 + κ 3 ± 2T 2 (κ 1 , κ 2 , κ 3 ) (4.40)
Hence, κ O 3 is also an integer and therefore, by the Antipodal relation, we have that the remaining curvatures

κ -1 = 2κ O 3 -κ 1 , κ -2 = 2κ O 3 -κ 2 and κ -3 = 2κ O 3 -κ 3
are also integers. The other value in Eq. (4.40) is the polytopal curvature

κ ′ O 3 of s f (B O 3 ) ⊂ Ω(B O 3
), which again must be an integer. We can then apply the same arguments to obtain that all the curvatures in s f (B O 3 ) are integers. By iterating the same process with other triples, we obtain that Ω(B O 3 ) is integral.

Now let us suppose that Ω(B O 3 ) is integral. Then T 2 (κ 1 , κ 2 , κ 3 ) = κ 1 κ 2 + κ 2 κ 3 + κ 1 κ 3 ∈ Z (4.41)
By combining the Antipodal relation with Corollary 4.2.1, we have that

2(κ 1 + κ 2 + κ 3 ) ± 2 2T 2 (κ 1 , κ 2 , κ 3 ) = 2κ O 3 = κ 1 + κ -1 ∈ Z Therefore, 2 2T 2 (κ 1 , κ 2 , κ 3 ) is an integer. Let us show it is an even integer. If there is m ∈ Z such that 2 2T 2 (κ 1 , κ 2 , κ 3 ) = 2m + 1, then we would have 2T 2 (κ 1 , κ 2 , κ 3 ) = m + 1 2 ⇔ 2T 2 (κ 1 , κ 2 , κ 3 ) = m 2 + m + 1 4 ̸ ∈ Z contradicting Eq. (4.41). Hence, 2 2T 2 (κ 1 , κ 2 , κ 3 ) is an even integer, so 2T 2 (κ 1 , κ 2 , κ 3 )
is also an integer.

We show in Figure 4.3, two integral octahedral Apollonian packings obtained from three initial disks whose curvatures satisfy the integrality condition of the previous proposition.

κ 1 + κ 3 = κ 2 + κ 4 (4.42)
By combining the corollary above with the Antipodal relation, the curvatures of all the disks in a cubical circle packing can be easily deduced from the curvatures of four disks which are not in a cyclic chain. Now we give the corresponding integrality condition, which can be proved similarly to the proof of Cor. 4.3.3. Then

Ω(B C 3 ) is integral if and only if κ 1 , κ 2 , κ 3 and 2C 2 (κ 1 , κ 2 , κ 3 ) are integers.

Icosahedral Apollonian packings

The curvatures of an icosahedral Apollonian packing cannot be in Z but in Z[φ].

We say that such a packing is φ-integral. The corresponding integrality condition for icosahedral Apollonian packings, which can be proved similarly to Cor. 4.3.3, is the following.

Corollary 4.3.6 (Integrality condition of icosahedral Apollonian packings). Let κ 1 , κ 2 , κ 3 be the curvatures of three mutually tangent disks of an icosahedral circle packing B I 3 . Then,

Ω(B I 3 ) is φ-integral if and only if κ 1 , κ 2 , κ 3 and T 2 (κ 1 , κ 2 , κ 3 ) are in Z[φ]. φ 2 φ 4 φ +3 5 φ +4 2 φ +2 4 φ -1 3 φ -2 6 φ 8 φ +3 7 φ +2 6 φ +2 8 φ +3 9 φ +4 6 φ +2 4 φ +3 3 φ +2 6 φ +4 12 φ +7 11 φ +6 10 φ +6 6 φ +4 6 φ +6 4 φ +3 6 φ -2 6 φ -4 8 φ -1 12 φ +3 11 φ +2 9 φ +4 14 φ +8 15 φ +6 16 φ +11 14 φ +8 9 φ +4 12 φ +3 14 φ +6 15 φ +10 14 φ +2 13 φ 17 φ +4 14 φ +8 18 φ +10 12 φ +7 10 φ +6 6 φ +4 12 φ +7 18 φ +10 13 φ +8 12 φ +7

Dodecahedral Apollonian packings

Like the cube, the dodecahedron is not simplicial. However, we can obtain from Eq. (4.38) a relation between the curvatures of the disks corresponding to the vertices lying in the same pentagonal face.

φ(κ i+1 -κ i ) = κ i+2 -κ i-1 (4.43)
where i runs cyclically from 1 to 5.

Corollary 4.3.8. Let κ i-1 , κ i , κ i+1 be the curvatures of three consecutive tangent disks in a dodecahedral circle packing

B D 3 . Then Ω(B D 3 ) is φ-integral if and only if κ i-1 , κ i , κ i+1 and -φ 2 κ 2 i + κ i κ i+1 + κ i κ i-1 + κ i-1 κ i+1 are in Z[φ].
-1 0 0

Introduction

In this chapter, we revisit the orthoplicial Apollonian packings introduced by Dias [START_REF] Dias | The Local-Global Principle for integral generalized Apollonian sphere packings[END_REF] and Nakamura [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF] as a three-dimensional generalization of the construction of Guettler and Mallows. An orthoplicial Apollonian packing can be obtained as follows. Consider a packing of 8 spheres whose tangency graph is the 1-skeleton of an orthoplex. For each subset of four pairwise tangent spheres, there is a unique dual sphere orthogonal to the four. Invert the whole configuration through every dual sphere, and then repeat this process indefinitely (Fig. 5.1). The structures introduced in this chapter will be used in Chapter 6.

The orthoplicial sphere packing

Nakamura defined a sphere packing to be orthoplicial if its tangency graph is the graph of an orthoplex [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF]. These packings were also studied by Sheydvasser as a packing arising from rational quaternion algebras [START_REF] Sheydvasser | Quaternion orders and sphere packings[END_REF]. They can also be obtained as a particular case of Boyd-Maxwell packing [START_REF] Chen | Lorentzian Coxeter systems and Boyd-Maxwell ball packings[END_REF] or a crystallographic sphere packing [START_REF] Kontorovich | Geometry and arithmetic of crystallographic sphere packings[END_REF]. In this section, we shall study orthoplicial sphere packings as polytopal sphere packings, i.e. as packings Möbius equivalent to the ball-arrangement projection of an edge-scribed orthoplex. As we show below, orthoplicial sphere packings, in the sense of Nakamura, are polytopal.

Lemma 5.2.1. All the sphere packings whose tangency graph is the graph of the orthoplex are polytopal.

Proof. Let B such a packing. The tangency relations give all the entries of Gram(B)

except for the entries corresponding to the inversive product of disjoint spheres. We shall prove that these inversive products must be all equal to -3. This would imply, by Th. 3.6.1, that we can reorder the spheres of B such that

Gram(B) = Gram(B O 4 ),
where B O 4 is the ball-arrangement projection of an edge-scribed orthoplex. Then, by Prop. 3.6.1, we would have that B and B O 4 are Möbius equivalent, and therefore B would be polytopal.

Let us give an antipodal labelling to

B = {b 1 , b 2 , b 3 , b 4 , b -1 , b -2 , b -3 , b -4 } so b i is always disjoint to b -i . For every 1 ≤ i < j ≤ 4, let B i,j ⊂ B the collection {b i , b j , b k , b l , b -i , b -j } where {i, j, k, l} = {1, 2, 3, 4}. Then, we have Gram(B i,j ) =            1 -1 -1 -1 λ i -1 -1 1 -1 -1 -1 λ j -1 -1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 λ i -1 -1 -1 1 -1 -1 λ j -1 -1 -1 1           
where λ i := ⟨b i , b -i ⟩. Since B i,j corresponds to a collection of 6 vectors of L 4,1 , then Gram(B i,j ) must be singular. Therefore, the entries λ 1 , λ 2 , λ 3 , λ 4 must verify the following conditions

λ i < -1 and (λ i -1)(λ j -1)(λ i + λ j + 6) = 0 for every 1 ≤ i < j ≤ 4 (5.1)
There is an only solution to the previous equations which is

λ 1 = λ 2 = λ 3 = λ 4 = -3
The first example of orthoplicial sphere packing considered by Nakamura and Dias is the standard orthoplicial sphere packing

[B O 4 ] 1 2 depicted in Fig. 5.2. b 3 b 4 b 1 b 2 b -2 b -1 b -4 b -3 Curvature Center / n Inversive coordinates b 1 0 (δ = 1) 0 0 1 0 0 1 1 1 b 2 0 (δ = 1) 0 0 -1 0 0 -1 1 1 b 3 1 1 1 0 1 1 0 0 1 b 4 1 -1 1 0 -1 1 0 0 1 b -1 2 0 0 -1/2 0 0 -1 -1 1 b -2 2 0 0 1/2 0 0 1 -1 1 b -3 1 -1 -1 0 -1 -1 0 0 1 b -4 1 1 -1 0 1 -1 0 0 1

Orthoplicial trinities

By polarity, the dual B * O 4 of an orthoplicial sphere packing is Möbius equivalent to the ball-arrangement projection of a ridge-scribed hypercube. Therefore, B * O 4 is not a packing. However, by alternating the vertices of the hypercube, we can split the dual in two orthoplicial sphere packings by a rotation of π 2 degree, one around the x-axis and the other around the y-axis, as shown in Fig. 5.3.

B * O 4 = B ′ O 4 ∪ B ′′ O 4 .

Apollonian groups of the orthoplicial sphere packing

In this context, the orthoplicial Apollonian group A(B O 4 ), introduced by Nakamura in [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF] and also by Dias in [START_REF] Dias | The Local-Global Principle for integral generalized Apollonian sphere packings[END_REF], corresponds to the Apollonian group of the standard orthoplicial sphere packing

V =     -1 1 1 1 1     , E = 1 2     1 1 -1 -1 0 1 1 1 1 0 -1 1 1 -1 0 -1 1 -1 1 0 0 0 0 0 2     , R =     1 1 -1 1 1     , F =     1 1 0 1 1 0 1     , S =     1 0 0 0 0 0 -1 0 -2 2 0 0 1 0 0 0 -2 0 -1 2 0 -2 0 -2 3     .
Proof. Let [B O 4 ] 1 2 be the standard orthoplicial sphere packing given in Figure 5.2. Let {r v , r e , r r , r f } be the fundamental generators of the symmetric group of [B O 4 ] 1 2 , with respect to the flag (v, e, r, f, O 4 ), where v = 1, e = 12, r = 123 and f = 1234. Then, {r v , r e , r r , r f } are represented by the following Möbius transformations:

• r v is the reflection on the plane {x = 0}.

• r e is the inversion on the sphere with center (-1, 1, -1) and radius 2.

• r r is the reflection on the plane {z = 0}.

• r f is the inversion on the sphere with center (0, 0, 1) and radius √ 2.

Since the orthoplex is regular, then SA([

B O 4 ] 1
S 1234 =     1 0 0 0 0 0 -1 0 -2 2 0 0 1 0 0 0 -2 0 -1 2 0 -2 0 -2 3     = S, S 1234 =     1 0 0 0 0 0 -1 -2 0 2 0 -2 -1 0 2 0 0 0 1 0 0 -2 -2 0 3     = FS 1234 F, S 1234 =     1 0 0 0 0 0 -1 2 0 2 0 2 -1 0 -2 0 0 0 1 0 0 -2 2 0 3     = RS 1234 R, S 1234 =     1 0 0 0 0 0 -1 0 2 2 0 0 1 0 0 0 2 0 -1 -2 0 -2 0 2 3     = FS 1234 F, S 1234 =     -1 0 0 -2 2 0 1 0 0 0 0 0 1 0 0 -2 0 0 -1 2 -2 0 0 -2 3     = ES 1234 E, S 1234 =     -1 0 -2 0 2 0 1 0 0 0 -2 0 -1 0 2 0 0 0 1 0 -2 0 -2 0 3     = FS 1234 F, S 1234 =     -1 0 2 0 2 0 1 0 0 0 2 0 -1 0 -2 0 0 0 1 0 -2 0 2 0 3     = RS 1234 R, S 1234 =     -1 0 0 2 2 0 1 0 0 0 0 0 1 0 0 2 0 0 -1 -2 -2 0 0 2 3     = FS 1234 F, S 1234 =     -1 0 0 2 -2 0 1 0 0 0 0 0 1 0 0 2 0 0 -1 2 2 0 0 -2 3     = VS 1234 V, S 1234 =     -1 0 2 0 -2 0 1 0 0 0 2 0 -1 0 2 0 0 0 1 0 2 0 -2 0 3     = FS 1234 F, S 1234 =     -1 0 -2 0 -2 0 1 0 0 0 -2 0 -1 0 -2 0 0 0 1 0 2 0 2 0 3     = RS 1234 R, S 1234 =     -1 0 0 -2 -2 0 1 0 0 0 0 0 1 0 0 -2 0 0 -1 -2 2 0 0 2 3     = FS 1234 F, S 1234 =     1 0 0 0 0 0 -1 0 2 -2 0 0 1 0 0 0 2 0 -1 2 0 2 0 -2 3     = ES 1234 E, S 1234 =     1 0 0 0 0 0 -1 2 0 -2 0 2 -1 0 2 0 0 0 1 0 0 2 -2 0 3     = FS 1234 F, S 1234 =     1 0 0 0 0 0 -1 -2 0 -2 0 -2 -1 0 -2 0 0 0 1 0 0 2 2 0 3     = RS 1234 R, S 1234 =     1 0 0 0 0 0 -1 0 -2 -2 0 0 1 0 0 0 -2 0 -1 -2 0 2 0 2 3     = FS 1234 F.
The previous representation of the orthoplicial Apollonian group in O ↑ 4,1 (Z) can be obtained by conjugating the matrix representation O ↑ W (Z) given by Nakamura in [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF], where W is the matrix of the inversive product in augmented curvature-center coordinates (see [START_REF] Lagarias | Beyond the Descartes circle theorem[END_REF]). Both representations satisfy the following relations: (R1) S 2 ijkl = 1 for every (i, j, k, l) = (±1, ±2, ±3, ±4).

(R2) (S ijkl S i ′ j ′ k ′ l ′ ) 2 = 1 if the labels ijkl and i ′ j ′ k ′ l ′ differ by only one letter.

Apollonian sections

Let B P be a polytopal d-ball packing and let A(B P ) and Ω(B P ) be the Apollonian group and the Apollonian cluster of B P . We say that a subset Σ ⊂ Ω(B P ) is an

Apollonian section of Ω(B P ) if there is subgroup Γ < A(B P ) and a subset X ⊆ B P such that Σ = Γ • X. Two Apollonian sections Σ = Γ • X and 2 , H T is the plane {y = 1}. After identifying H T with R 2 , we obtain that

Σ ′ = Γ ′ • X ′ of
X T ∩ H T := {b 1 ∩ H T , b 2 ∩ H T , b 3 ∩ H T , b 4 ∩ H T }
becomes a tetrahedral circle packing B T 3 . Moreover, the restriction of Γ T on H T induces an isomorphism between Γ T and A(B T 3 ) given by

ϕ T : Γ T -→ A(B T 3 ) s b -→ s b∩H T
where s b denotes the inversion on b. On the other hand, we can construct a bijection

ψ T : Σ T -→ Ω(B T 3 ) g • b -→ (g • b) ∩ H T
By inspection, one can check that for every ijkl ∈ {1234, 1234, 1234, 1234} and for every m = 1, 2, 3, 4, we have

ψ T (s ijkl • b m ) = (s ijkl • b m ) ∩ H T = ϕ T (s ijkl ) • (b m ∩ H T ) = ϕ T (s ijkl ) • ψ T (b m )
which gives the equivariance of ψ T . Therefore, Σ T is algebraically equivalent to the tetrahedral Apollonian packing Ω(B T 3 ).

O := Γ O • X O ⊂ Ω([B O 4 ] 1 2 )
where The bijections ψ T and ψ O described above, both preserve the inversive product, which is not the case for ψ C . Indeed, spheres which are tangent in the cubical Apollonian section may correspond to disjoint disks in the cubical Apollonian packing.

Γ O =
1 1 -1/ √ 2 ( √ 2 + 1) ( 1 1 1) 1/ √ 2 ( 1 1 1 1 √ 2) b 2 1 + 1/ √ 2 ( √ 2 -1) ( 1 -1 1) 1/ √ 2 ( 1 -1 1 -1 √ 2) b 3 1 -1/ √ 2 ( √ 2 + 1) ( 1 -1 -1) 1/ √ 2 ( 1 -1 -1 1 √ 2) b 4 1 + 1/ √ 2 ( √ 2 -1) ( 1 1 -1) 1/ √ 2 ( 1 1 -1 -1 √ 2) b -1 1 + 1/ √ 2 ( √ 2 -1) (-1 -1 -1) 1/ √ 2 (-1 -1 -1 -1 √ 2) b -2 1 -1/ √ 2 ( √ 2 + 1) (-1 1 -1) 1/ √ 2 (-1 1 -1 1 √ 2) b -3 1 + 1/ √ 2 ( √ 2 -1) (-1 1 1) 1/ √ 2 (-1 1 1 -1 √ 2) b -4 1 -1/ √ 2 ( √ 2 + 1) (-1 1 -1) 1/ √ 2 (-1 -1 1 1 √ 2)
To see this, we might consider the cubical Apollonian packing Ω(B On the arithmetic side, the bijections ψ T and ψ O preserve the curvatures, contrarily to ψ C . However, by composing ψ C with a rescaling of R 2 of factor √ 2 we obtain a bijection that does preserve the curvatures. These properties will be important in the next section in order to construct integral orthoplicial Apollonian packings containing an Apollonian section with prescribed curvatures.

Construction of orthoplicial Apollonian packings containing a given integral section

In this section, we prove the main result of this chapter. by applying a translation of R 3 of vector (-1, 0, 0), so H T becomes the plane given by {x = 0}. Let Σ T = Γ T •X T be the corresponding tetrahedral section of Ω(B O 4 ) and let B T 3 be the tetrahedral circle packing given by ψ T (X T ). In inversive coordinates,

the bijection ψ -1 T : Ω(B T 3 ) → Σ T ⊂ Ω(B O 4
) is given by

ψ -1 T : Ω(B T 3 ) -→ Σ T i(b) -→ 0 i(b)
Let B ′ T 3 be any tetrahedral circle packing. Since the tetrahedron is Möbius unique, there exists µ ∈ Möb( R 2 ) such that µ

• B T 3 = B ′ T 3 . Therefore, µ • Ω(B T 3 ) = Ω(B ′ T 3 ). Let M ∈ O ↑
3,1 (R) be the matrix corresponding to µ and let μ ∈ Möb( R 3 ) be the Möbius transformation corresponding to the matrix

1 0 T 4 0 4 M ∈ O ↑ 4,1 (R)
where 0 4 is the null column-matrix of size 4. We have that Σ ′ T := µ • Σ T is a tetrahedral section of the orthoplicial Apollonian packing Ω(B ′ O 4 ) = µ • Ω(B O 4 ). We define the bijection

ψ T := µ • ψ T • µ -1 mapping Σ ′ T to Ω(B ′ T 3 ). Let us show that Σ ′ T preserves curvatures. For every disk b ∈ Ω(B ′ O 3 ), we have κ( ψ -1 T (b)) = κ( µ • ψ -1 T • µ -1 (b)) = k 5 1 0 T
ψ -1 C : B C 3 -→ B ↑x O 4 i(b) -→ 1 √ 2 ε(b) i(b) ϕ -1 C : A(B C 3 ) → Γ C A → 1 0 T 4 0 4 A (5.
3)

The equivariance of ψ C allows us to extend

ψ -1 C : Ω(B C 3 ) → Σ C by ψ -1 C (g • b) = ϕ -1 C (g) • ψ -1 C (b) (5.4)
for every g ∈ A(B C 3 ) and every b ∈ B C 3 . Let B ′ C 3 be any cubical circle packing. As before, the Möbius uniqueness of the cube implies that there is µ ∈ Möb( R 2 ) such that µ • Ω(B C 3 ) = Ω(B ′ C 3 ). We define µ ∈ Möb( R 3 ) as above in the tetrahedral case.

Let µ = λ 1 √ 2 • µ ∈ Möb( R 3 ) where λ 1 √ 2
is the rescaling of R 3 with factor 1 √ 2 . We have that Σ ′ C := µ • Σ C is a cubical Apollonian section of the orthoplicial Apollonian packing Ω(B ′ O 4 ) := µ • Ω(B ↑x O 4 ). We define the bijection

ψ C = µ • ψ C • µ -1 which maps Σ ′ C → Ω(B ′ C 3 )
. Again, we shall prove that ψ C preserves curvatures. First, we notice that for any disk b ∈ Ω(B ′ C 3 ), there is

g ′ ∈ A(B ′ C 3 ) and b ′ ∈ B ′ C 3 such that b = g ′ • b ′ , by 102
κ ψ -1 C (b) = κ ψ -1 C (g ′ • b ′ ) = κ µ • ψ -1 C • µ -1 (g ′ • b ′ ) = κ λ 1 √ 2 • µ • ψ -1 C • µ -1 (g ′ • b ′ ) = √ 2κ µ • ψ -1 C • µ -1 (µgµ -1 ) • (b ′ ) where g = µ -1 g ′ µ ∈ A(B C 3 ) = √ 2κ µ • ϕ -1 C (g) • ψ -1 C • µ -1 (b ′ ) by (5.4) = √ 2k 5 1 0 T 4 0 4 M 1 0 T 4 0 4 A 1 √ 2 ε(µ -1 (b ′ )) i(µ -1 (b ′ ))
where A is the matrix of g

= k 5 ε(µ -1 (b ′ )) MAi(µ -1 (b ′ )) = k 4 MAi(µ -1 (b ′ )) = k 4 MAM -1 i(b ′ ) = κ(µgµ -1 • b ′ ) = κ(g ′ • b ′ ) = κ(b)
Now we suppose that Ω(B ′ T 3 ), Ω(B ′ O 3 ) and Ω(B ′ C 3 ) are integral. We shall show that in the three cases we can find four pairwise tangent spheres of B ′ O 4 with curvatures κ 1 , κ 2 , κ 3 and κ 4 satisfying that T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) ∈ Z. The integrality of Ω(B ′ O 4 ) then follows from Prop 5.3.1.

(Tetrahedral section) Let κ 1 , κ 2 , κ 3 , κ 4 be the curvatures of the four disks of B ′ T 3 . By the Descartes' Theorem, we have that

T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) = 0 Since ψ -1
T preserves curvatures, then κ 1 , κ 2 , κ 3 , κ 4 are the curvatures of four pairwise tangent spheres of B ′ O 4 satisfying that T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) ∈ Z. (Octahedral section) Let κ 1 , κ 2 , κ 3 , κ -1 , κ -2 , κ -3 be the curvatures of the six disks of B ′ O 3 under an antipodal labelling, so κ i and κ -i are the curvatures of non-tangent disks. By (4.27) , we have

(κ 1 -κ O 3 ) 2 + (κ 2 -κ O 3 ) 2 + (κ 3 -κ O 3 ) 2 = 2κ 2 O 3
(5.5)

Apollonian sections 103

where κ O 3 = 1 2 (κ i + κ -i ) for every i = 1, 2, 3. Since ψ -1 O preserves curvatures, then κ 1 , κ 2 , κ 3 , κ -1 , κ -2 ,
(κ 1 -κ O 4 ) 2 + (κ 2 -κ O 4 ) 2 + (κ 3 -κ O 4 ) 2 + (κ 4 -κ O 4 ) 2 = 2κ 2 O 4 ⇔ (κ 1 -κ O 3 ) 2 + (κ 2 -κ O 3 ) 2 + (κ 3 -κ O 3 ) 2 + (κ 4 -κ O 3 ) 2 = 2κ 2 O 3 ⇔ κ 4 = κ O 3 Therefore, T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) = T 3 (κ 1 , κ 2 , κ 3 , κ O 3 ) = 1 2 (κ 1 + κ 2 + κ 3 + κ O 3 ) 2 -2 κ 2 1 + κ 2 2 + κ 2 3 + κ 2 O 3 = 1 2 2(κ 1 κ 2 + κ 2 κ 3 + κ 1 κ 3 ) + 2κ 2 O 3 -(κ 1 -κ O 3 ) 2 -(κ 2 -κ O 3 ) 2 -(κ 3 -κ O 3 ) 2 = 1 2 2(κ 1 κ 2 + κ 2 κ 3 + κ 1 κ 3 ) by (5.5) = 1 2 2T 2 (κ 1 , κ 2 , κ 3 ) Since Ω(B ′ O 3 ) is integral, then, 2T 2 (κ 1 , κ 2 , κ 3
) is an even integer. Moreover, by Cor. 4.3.3, 2T 2 (κ 1 , κ 2 , κ 3 ) is also an even integer. Consequently, T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) ∈ Z.

κ 1 + κ 3 = κ 2 + κ 4
Since ψ -1 C preserves curvatures, then κ 1 , κ 2 , κ 3 , κ 4 are the curvatures of four pairwise tangent spheres of B O 4 . Then, we have

T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) = T 3 (κ 1 , κ 2 , κ 3 , κ 1 + κ 3 -κ 2 ) = κ 1 κ 2 + κ 2 κ 3 + κ 1 κ 3 -κ 2 2 = 2C 3 (κ 1 , κ 2 , κ 3 )
The integrality of Ω(B ′ C 3 ) and Cor. 4.3.5 give that T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) ∈ Z.

Chapter 5. Apollonian sections of the orthoplicial Apollonian packing

The previous proof is constructive and gives a method to obtain integral orthoplicial Apollonian packings containing a given tetrahedral, octahedral or cubical section. The three orthoplicial Apollonian packings shown in Table 5.1 were obtained by this method.

Platonic AP Orthoplicial AP Apollonian section -3 4 In this last chapter before the conclusion, we shall take advantage of the fractal structure of the orthoplicial Apollonian sphere packing to construct necklace representations of links with few spheres. After showing that any link admits an orthoplicial necklace representation, we shall improve the upper bound on the ball number given in Theorem 2.1.1 for rational links. We conclude with an unexpected relation between the arithmetical properties of rational links and the inversive coordinates of a tangency point of two spheres in an orthoplicial necklace representation. The latter puts forward a connection between continued fractions, a cubic Apollonian packing and a Diophantine equation.

Orthoplicial necklace representations

We say that a necklace representation of a link is orthoplicial if it is contained in an orthoplicial Apollonian packing. In the following result, we show that any link admits an orthoplicial necklace representation. Theorem 6.2.1. Any link admits a necklace representation in any orthoplicial Apollonian sphere packing.

Proof. Let L be a link. By Th. 1.2.3, there is an n-braid γ such that its closure is isotopically equivalent to L. We shall show that the closure of γ can be constructed geometrically in the carrier of the orthoplicial Apollonian sphere packing [B O 4 ] 1 2 shown in the Figure 5.2.

First, we construct the Apollonian section Σ

12 := Γ 12 • [B O 4 ] 1 2 ⊂ Ω([B O 4 ] 1
2 ) where Γ 12 := ⟨s 1234 , s 1234 , s 1234 , s 1234 ⟩ (6.1)

We have that Σ 12 is an infinite sphere packing whose carrier is a square-grid with two vertices, one above the other, at the center of each square. Therefore, via squaregrid diagrams, we can construct a polygonal path in the carrier of Σ 12 equivalent to the closure of γ by using the vertices at the center of the squares for the crossings (see Fig. The method used in the previous proof can be used to give an upper bound of the ball number of a link in terms of its braid length. However, this method only improves the upper bound of Theorem 2.1.1 for links whose braid index is 2. The closure of a 2-braid with n crossings is equivalent to the rational link C(n).

In the next section, we shall extend the improved upper bound of Corollary 6.2.1 to every rational link with the help of a cubical Apollonian section of an orthoplicial Apollonian packing.

Algebraic links in the cubical Apollonian section

In order to simplify the latter computations, instead of working with the orthoplicial sphere packing B ↑x O 4 where the cubical Apollonian section was defined in the proof of Prop. We call these edges the diagonal edges.

1 b -4 b 3 b -2 b -1 b 4 b -3 b 2 x y Curvature Center Inversive coordinates b 1 1 -1/ √ 2 ( √ 2 + 1) ( 1 1 1) 1/ √ 2 ( 1 1 1 1 √ 2) b 2 1 + 1/ √ 2 ( √ 2 -1) (-1 1 1) 1/ √ 2 (-1 1 1 -1 √ 2) b 3 1 -1/ √ 2 ( √ 2 + 1) (-1 -1 1) 1/ √ 2 (-1 -1 1 1 √ 2) b 4 1 + 1/ √ 2 ( √ 2 -1) ( 1 -1 1) 1/ √ 2 ( 1 -1 1 -1 √ 2) b -1 1 + 1/ √ 2 ( √ 2 -1) (-1 -1 -1) 1/ √ 2 (-1 -1 -1 -1 √ 2) b -2 1 -1/ √ 2 ( √ 2 + 1) ( 1 -1 -1) 1/ √ 2 ( 1 -1 -1 1 √ 2) b -3 1 + 1/ √ 2 ( √ 2 -1) ( 1 1 -1) 1/ √ 2 ( 1 1 -1 -1 √ 2) b -4 1 -1/ √ 2 ( √ 2 + 1) (-1 1 -1) 1/ √ 2 (-1 1 -1 1 √ 2)

Orthocubical shifts

We consider the dual

B * C 3 = {d 1 , d 2 , d 3 , d -1 , d -2 , d -3
} with the antipodal labelling depicted in Figure 6.6 and the elements {r 12 , r 13 , r 23 , r -13 , r -23 , r -33 } ⊂ Sym(B * C 3 ) where r ij denotes the inversion which exchanges the disks d i with d j , d -i with d -j , and fixes the rest. In B * C 3 we have that r 12 corresponds to the reflection on the line {x = y}, r ±i3 is the inversion on the circle centered at ±e i and radius √ 2, for i = 1, 2, and r -33 is the inversion on the unit circle centered at the origin (see Fig. Since Sym(B C 3 ) = Sym(B * C 3 ), the elements {r 12 , r 13 , r 23 , r -13 , r -23 , r -33 } belong also to the symmetrized Apollonian group SA(B C 3 ). We define the cubical shifts as the elements {µ x , µ y , µ z , µ -x , µ -y , µ -z } ⊂ SA(B C 3 ) given by

µ ±x := s ±1 r ±13 µ ±y := s ±2 r ±23 µ ±z := s ±3 r -33 (6.2)
where s i denotes the inversion on the disk d i ∈ B * C 3 . In the Figure 6.7, we show the action of the cubical shifts on the carrier of B C 3 . We notice that µ ±x and µ ±y preserve and µ ±z reverses the z-coloring, and the subgroup generated by {µ x , µ y , µ -x , µ -y , µ -z } is free.

An upper bound on the ball number of rational links

We shall present here a discrete analogue of algebraic links by using the algebraic structure given by the Apollonian groups of the orthocubical sphere packing.

First, we define an orthocubical path γ as a polygonal curve in the carrier of Σ C connecting the centers of two spheres of Σ C . Orthocubical paths shall be represented by cubical diagrams (see Figure 6.5) which are obtained by adding the diagonal edges to the carrier of the corresponding cubical Apollonian packing. We recall that the over/under information for the crossings in the cubical diagrams is given by the color of the vertices in the diagonal edges (black=over/white=under). We shall also encode an orthocubical path by a vector (c i 1 , • • • , c in ) which denotes the centers of the spheres in the linear order induced by γ. The endpoints of γ are the centers c i 1 and c in and the length of γ, denoted by |γ|, is the number n counting the centers in γ.

Since we shall consider unoriented paths, and the concatenation of two paths gives another path, vectors encoding orthocubical paths must be considered modulo the following relations:

(i) (Symmetry) (c i 1 , . . . , c in ) = (c in , . . . , c i 1 ) (ii) (Concatenation) {(c i , • • • , c j ), (c j , • • • , c k )} = {(c i , • • • , c j , • • • , c k )}
Let T be the tetrahedron T := conv({c 1 , c -2 , c 3 , c -4 }) where c i is the center of the sphere b i ∈ B ↑z O 4 . We define an orthocubical tangle as a 2-tangle (T , t ) where t is a collection {γ 1 , γ 2 , . . . , γ m } of m ≥ 2 disjoint orthocubical paths contained in T satisfying that the endpoints of γ 1 and γ 2 are {c 1 , c -2 , c 3 , c -4 } and the rest of the orthocubical paths are closed. The length of t , denoted by | t |, is the sum of the lengths of the orthocubical paths in t . We denote by t 0 , t 1 and t ∞ the elementary orthocubical tangles depicted in Fig. 6.8.

c-3 c2 c4 c-1 t H + O t c 1 c -4 c -2 c 3 c-3 c2 c4 c-1 t H - O t
Figure 6.9: Orthocubical unary operations of tangles.

The orthocubical tangle closures are given by:

Algebraic links in the cubical Apollonian section

115

-The orthocubical numerator

N O t := t ∪ {(c 1 , µ z (c 1 ), µ z (c -4 ), c -4 ), (c 3 , µ z (c 3 ), µ z (c -2 ), c -2 )}.
-The orthocubical denominator

D O t := t ∪ {(c 1 , µ z (c 1 ), µ z (c -2 ), c -2 ), (c 3 , µ z (c 3 ), µ z (c -4 ), c -4 )}. t t c 1 c -4 c -2 c 3 t N O t µ z (c 1 ) µ z (c -4 ) µ z (c -2 ) µ z (c 3 ) c 1 c -4 c -2 c 3 t D O t µ z (c 1 ) µ z (c -4 ) µ z (c -2 ) µ z (c 3 ) c 1 c -4 c -2 c 3
rational tangle t (a 1 , • • • , a n ) equivalent to the rational tangle t(a 1 , • • • , a n ) by t (a 1 , • • • , a n ) := M O (a 1 ) • • • M O (a n ) t ∞ where M O (a i ) := (H + O ) a i F O if a i ≥ 0 (H - O ) a i F O if a i < 0 (6.3)
By combining the orthocubical rational tangles with the orthocubical operations Proof. Let L be a link equivalent to the closure of rational tangle t p/q , where p/q ∈ Q∪ {∞}. Let a 1 + 1

(+ O , F O , N O , D O )
•••+ 1 n
be a continued fraction expansion of p/q where all the coefficients have the same sign (a 1 might be zero). Then, L is equivalent to the rational link C(a 1 , . . . , a n ). If n = 1, then the upper bound was given in the Corollary 6.2.1. Let us suppose that n ≥ 2. Since the ball number of a link and its mirror is the same, we can suppose that a 1 ≥ 0 and a i ≥ 1 for 2 ≤ i ≤ n. By the equivalence between orthocubical rational tangles and rational tangles, we have that L is equivalent to the orthocubical tangle closure N O t where Finally, since a i ≥ 1, the orthocubical paths (c 1 , c -4 ) and (c 3 , c -2 ) are not in t ′ . Therefore, we can use these paths to close t ′ , and in this way obtain a necklace representation of L with 4cr(L) spheres.

t = (H + O ) a 1 F O • • • (H + O ) an F O t ∞ = (H + O ) a 1 F O • • • (H + O ) an t 0 = (H + O ) a 1 F O • • • (H + O ) an-1 ( t 1 + t 0 ) ≃ (H + O ) a 1 F O • • • (H + O ) an-1 t 1 since t 1 + t 0 ≃ t 1 =: t ′ It follows
The improved upper bound can be extended to algebraic links admitting an alternating algebraic presentation of sums and flips of rational tangles. In the nonalternating case, the number of spheres in the orthocubical necklace representation may be reduced to less than 4 times the crossing number. The first non-trivial example that we have found satisfying this property, is precisely the Pretzel knot P (3, -2, 3) (Fig. 1.12). This knot is not alternating [START_REF] Cromwell | Knots and Links[END_REF], it admits an orthocubical necklace representation with 28 spheres (Fig. 6.11), and its crossing number is 8 [START_REF] Cromwell | Knots and Links[END_REF]. However, it becomes more difficult to establish a relation with the crossing number in the non-alternating case since, in general, the crossing number does not correspond to the sum of the crossings of its rational factors.

A geometric interpretation of continued fractions

In the words of Karpenkov, the ubiquity of continued fractions is due to the simple operations that compose a continued fraction: sum a + b and inversion 1/a [START_REF] Karpenkov | Geometry of continued fractions[END_REF]. When a mathematical object admits an analogue of these two operations, there is probably a way to connect it with a rational number via continued fractions.

In [START_REF] Chaubey | The dynamics of super-Apollonian continued fractions[END_REF], Chaubey, Fuchs, Hines and Stange related continued fractions with the tetrahedral Apollonian packings and Pythagorean triples by using the generators of the Super Apollonian group introduced in [START_REF] Graham | Apollonian circle packings: Geometry and group theory i. the Apollonian group[END_REF] which is provided by the Lorentzian model of the space of disks. In this last section, we shall show that the continued fraction involved in the construction of a rational tangle can be read from the inversive coordinates of a tangency point in the orthocubical version of Conway's algorithm. This result might be related with another work of Stange [START_REF] Stange | The sensual Apollonian circle packing[END_REF], where the author computed the complex coordinates of the tangency points of certain disks in the standard tetrahedral Apollonian packing by relating Apollonian packings with another work of Conway and Fung on quadratic forms [START_REF] Conway | The sensual (quadratic) form[END_REF].

118 Chapter 6. Orthoplicial Apollonian packings and rational links Definition 6.4.1. Let t p/q be a rational tangle with positive continued fraction expansion

a 1 + 1 •••+ 1 an = p q .
We define:

-The first edge of t p/q as the segment e p/q of t (a 1 , • • • , a n ) containing the center c 1 .

-The first tangency point of t p/q is the tangency point η p/q of the two spheres corresponding to the vertices of e p/q in the orthocubical sphere packing.

We shall represent both e p/q and η p/q by their respective projections e p/q and η p/q in the cubical diagrams as is shown in Fig. 6.12. Since the vertices of e p/q have different colors under the z-coloring, we have that η p/q lies in the plane {z = 0}. This implies that the Cartesian coordinates of η p/q can be computed from the Cartesian coordinates of η p/q by adding 0 in the third coordinate. The inversive coordinates of the latter will allow us to find the fraction p/q from the geometric construction of the corresponding orthocubical tangle when the fraction is positive. Theorem 6.4.1. For every coprime integers p, q with q ≥ 0 and p ≤ 1, we have

i(η p/q ) =       p 2 q 2 (p -q) 2 √ 2(p 2 -pq + q 2 )       (6.4)
Proof. The positiveness of p and q implies that we can find a positive continued fraction expansion a 1 + 1

•••+ 1 an = p/q with a 1 ≥ 0 and a i ≥ 1 for every i > 1. Let η p/q and η ∞ be the first tangency points of t p/q and t ∞ respectively. By the definition of the orthocubical Conway's algorithm (c.f. 6.3), we have that

e p/q = (c 1 , m a 1 • • • m an (c -2 ))
where m a i := µ a i x r 24 . Recall that µ x is an orthocubical shift and r 24 ∈ Sym(B ↑z O 4 ). Therefore,

η p/q = m a 1 • • • m an (η ∞ ) ⇒ η p/q = m a 1 • • • m an (η ∞ )
where m a i = µ a i

x r 12 = (s 1 r 13 ) a i r 12 . The elements s 1 , r 13 and r 12 are the elements of Sym(B * C 3 ) described at the beginning of Section 6.3.1. The inversive coordinates of η ∞ and the matrices representing s 1 , r 13 and r 12 can be computed with Equations

M(k) =       0 1 -k 2 -k(k + 2) √ 2k(k + 1) 1 0 0 0 0 -k(k -2) 1 -k 2 √ 2k(k -1) 0 - √ 2k(k -1) - √ 2k(k + 1) 2k 2 + 1      
We prove the statement of the theorem by induction on the number of coefficients n in the fraction expansion of p/q. For n = 1 (so p = a 1 and q = 1) we have (ra 1 + s -r) 2 √ 2((ra 1 + s) 2 -r(ra

M(a 1 )       1 0 1 √ 2       =      
1 + s) + r 2 )      
We finally notice that

ra 1 + s r = a 1 + s r = a 1 + 1 r/s = a 1 + 1 a 2 + 1 •••+ 1 an =
p q so the theorem holds for n.

A geometric interpretation of continued fractions 121

The following two corollaries follow from the previous theorem and the formula giving the inversive coordinates of a point. obtained as the intersection described in Corollary 6.4.1. For every p ≥ q > 1, the fraction p/q is printed at the center of the white disk corresponding to the vertex e p/q distinct from c 1 .

Corollary 6.4.2. The Diophantine equation x 4 + y 4 + z 4 = 2t 2 has an infinite number of primitive solutions.

Proof. Let p and q be two coprime positive integers. Since points of R 2 correspond to light-like vectors of L 3,1 , we have ⟨η p/q , η p/q ⟩ = i(η p/q ) T Q 4 i(η p/q ) = 0 (6.5)

where Q 4 = diag(1, 1, 1, -1). By combining Equations (6.4) and (6.5), we obtain a primitive solution of the Diophantine equation by setting

x = p, y = q, z = p -q and t = p 2 -pq + q 2 (6.6)

Conclusions: current and future work

We summarize here the main results of this thesis and we discuss the current work and future perspectives.

Möbius uniqueness

In Chapter 1, we have discuss the notion of Möbius uniqueness for planar graphs, which arises from the uniqueness of planar embeddings of graphs under Möbius transformations involved in the KAT theorem. We then extend this notion to edgescribable polytopes in Chapter 3. We studied the Möbius uniqueness for most of regular polytopes, and used the derived results in the Chapters 4, 5 and 6. In particular, we have shown that the family of d-simplices, (d + 1)-cross polytopes, (d + 1)cube for every d ≥ 2, and also the 24-cell, are Möbius unique (Corollaries 3.6.1, 3.6.4, 3.6.5). For the non-regular case, we have shown that the non-regular 2-polytopes are not Möbius unique (Corollary 3.6.2), contrarily to the 3-polytopes, which, as a consequence of the Midsphere Theorem, are all Möbius unique. Unfortunately, for d ≥ 4, there is no analogue of the Midsphere Theorem, but we believe that all edge-scribable 4-polytopes are Möbius unique. In light of all this, we propose the following conjecture, see [START_REF] Ramírez Alfonsín | A polytopal generalization of Apollonian packings and Descartes' theorem[END_REF].

Conjecture 1. For every d ≥ 3, all edge-scribable d-polytopes are Möbius unique.

In view of the family of polytopes that we found to be Möbius unique, a natural next step would be to show the validity of the Conjecture 1 for regular polytopes.

Question 1. Are the 600-cell and the 120-cell Möbius unique?

After discussing Möbius uniqueness, we introduced a related spectral invariant for edge-scribable polytopes that are Möbius unique that we have called Möbius spectra. On the other hand, the well-known Steinitz's Theorem [START_REF] Steinitz | Über isoperimetrische probleme bei konvexen polyedern[END_REF] states that the graph of a polyhedron is a 3-connected simple planar graph. Such graphs are usually called polyhedral graphs. Since all the polyhedra are edge-scribable and Möbius In Chapter 5, we applied the techniques used in the previous chapters on the orthoplicial Apollonian packings studied by Dias [START_REF] Dias | The Local-Global Principle for integral generalized Apollonian sphere packings[END_REF] and Nakamura [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF]. Moreover, we have introduced the notion of Apollonian section as an algebraic generalization of geometric cross-sections. We found that the tetrahedral, octahedral and cubical Apollonian packings studied in Chapter 4 can be found as slices of the orthoplicial Apollonian packing. The main point of this result is that the inclusion between the different Apollonian packings, is not only geometrically obtained (or algebraically regarding Apollonian groups) but also arithmetically, that is, every integral tetrahedral, octahedral and cubical Apollonian packing can be obtained by intersecting an integral orthoplicial Apollonian packing with a plane (Th. 5.3.1).

An interesting open problem in number theory on Apollonian circle packings is the Local-to-Global conjecture of Graham et al. [START_REF] Graham | Apollonian circle packings: number theory[END_REF], and improved by Fuchs and Sanden [START_REF] Fuchs | Some experiments with integral Apollonian circle packings[END_REF]. This conjecture says that in any integral tetrahedral Apollonian circle packing Ω(B T 3 ), every integer avoiding certain restrictions modulo 24 must appear as the curvature of Ω(B T 3 ). Zhang [START_REF] Zhang | On the local-global principle for integral Apollonian 3-circle packings[END_REF], Kontorovich [START_REF] Kontorovich | The local-global principle for integral soddy sphere packings[END_REF], Dias [START_REF] Dias | The Local-Global Principle for integral generalized Apollonian sphere packings[END_REF] and Nakamura [START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF] proved the analogue of the Local-to-Global conjecture for octahedral, simplicial and orthoplicial Apollonian packings, respectively. We believe that Th. 5.3.1 may be helpful to better understand the original Local-to-Global conjecture in the tetrahedral case.

The choice of studying orthoplicial Apollonian packings in Chapter 5, among the Apollonian packings based on regular 4-polytopes like the simplex or the hypercube, was not arbitrary. We decided to explore this case from the results on the ball number that are developed in the last chapter of this thesis. Nevertheless, we believe that it would be interesting to study the Apollonian groups, packings and sections of the other regular 4-polytopes. On the integrality side, numerical experiments on the Apollonian packing of the 24-cell suggested the following two conjectures, see [START_REF] Rasskin | Regular polytopes, sphere packings and Apollonian sections[END_REF].

Conjecture 2. Let B R 4 be a standard polytopal sphere packing, obtained by rescaling a 1-CBP projection of the 24-cell by a factor of 1 √ 3 . Then, the set of curvatures of its Apollonian packing Ω(B R 4 ) is N.

Necklace representations and the ball number

In Chapters 2 and 6, we presented two methods of construction of necklace representations of knots and links with relatively few spheres. The first method is based on the KAT theorem and yields the first upper bound on the ball number of nonsplittable and nontrivial links, which is 5 times the crossing number. The second method uses the fractal structure of the orthoplicial Apollonian packing and improves the previous upper bound for rational links to 4 times the crossing number. In addition, all the necklace representations that we found with less than 4 times the crossing number, such as the Pretzel knot P (3, -2, 3) (Fig. 6.11), were for non-alternating knots or links. The reason is that non-alternating crossings can be done with fewer spheres. Hence, we believe that the second upper bound is actually tight for alternating links. This leads to the following conjecture, see [START_REF] Ramírez Alfonsín | Ball packings for links[END_REF].

Conjecture 5. The ball number of an alternating link L is 4cr(L).

Regarding lower bounds on the ball number, we were not able to find any nontrivial lower bound other than the one provided by the stick number. On this line, Maehara compared in [START_REF] Maehara | Pearl numbers versus stick numbers for knots[END_REF] the stick number against the pearl number (defined similarly as the ball number but with equal size spheres) and gave non-trivial lower bounds on the pearl number of knots.

We have chosen the orthoplex for the construction of necklace representations for two main reasons. Firstly, it is the only regular 4-polytope whose standard Apollonian packing contains clearly a square-grid in the carrier (Fig. 6.2). This is due to the fact that the edge-link of the orthoplex is a square, unlike the other regular 4polytopes, whose edge-link is a triangle (4-simplex, hypercube, 24-cell and 120-cell) or a pentagon (600-cell). The latter was used to prove that the orthoplicial Apollonian packing has a structure complex enough to contain any link in its carrier (Th. 6.2.1). Secondly, such structure is manageable, since it can be seen as the action of a reflection group with relatively few generators. This was helpful in order to find the good elements in the corresponding symmetrized Apollonian group to produce the desired constructions. We believe that analogous methods, for the other regular 4-polytopes, should be considered. For instance, we have started to explore necklace representations of three interesting links, that can be found in a CBP projection Conlusions: current and future work of the orthoplex O 4 , 24-cell R 4 and 600-cell I 4 , respectively. For the construction, we use a well-known fact, that the vertices of three 4-polytopes O 4 , R 4 and I 4 , can be equally partitioned into 2, 4 and 12 subsets of 4, 6 and 10 vertices, respectively, where each subset is contained in a fiber of the Hopf fibration (see [START_REF] Coxeter | Regular Polytopes. Dover books on advanced mathematics[END_REF][START_REF] Banchoff | Torus Decompostions of Regular Polytopes in 4-space[END_REF][START_REF] Schleimer | Puzzling the 120-cell[END_REF]). On the other hand, it is also known that the link obtained from selecting n fibers in the Hopf fibration is a link of n components satisfying that any two form a Hopf link [START_REF] Hopf | Über die abbildungen der dreidimensionalen sphäre auf die kugelfläche[END_REF].

Let us denote this link by Hf (n). Therefore, the latter partitions produce necklace representations of the links Hf (2), Hf (4) and Hf (12), respectively. It seems that the number of spheres obtained from this construction, with respect to the crossing number, is much smaller than the constructions presented here. We show in Figures Finally, we would like to point out an important difference between the different methods of construction of necklaces. Since the algorithms of circle packings, induced by the KAT theorem, are an infinite limit process, the inversive coordinates obtained by the method given in Chapter 2 have approximate values. On the contrary, the orthoplicial necklace representations obtained by either closing orthoplicial braids or the orthocubical Conway's algorithm, give algebraic values for the inversive coordinates of the spheres. We believe that this can be interesting for the study of the hyperbolic volume of hyperbolic 3-manifolds [START_REF] Gabai | Hyperbolic 3-manifolds of low cusp volume[END_REF].

Further knotted constructions

Beyond knots and links, we are also applying the techniques developed in this thesis in the realization of other spatial constructions. We recall that a Seifert surface of a link is an oriented surface containing the link as a boundary. By applying the with polytopes, Apollonian packings and knots. Polytopack will eventually be made available for researchers and other users interested in these topics.

Below, we give a few examples of the tools that have been implemented in Polytopack. We end by illustrating some mathematical applications of Polytopack that have not been treated in this dissertation.

  3, nous faisons le lien entre les empilements et les polytopes. Pour cela, nous voyons d'abord les définitions classiques associées aux polytopes et le modèle projectif de l'espace de boules. Dans ce modèle, une boule d-dimensionnelle correspond à un point dans l'espace euclidien de dimension d + 1 ayant une norme strictement plus grande que 1. De manière analogue, un empilement de boules ddimensionnelles correspond à un polytope de dimension d + 1 vérifiant certaines propriétés géométriques. Les empilements issus de cette correspondance seront appelés polytopaux. Nous étudions ensuite l'unicité sous transformations de Möbius et la dualité des empilements polytopaux. Cette dernière permet de bien généraliser les empilements apolloniens dans d'autres combinatoires et en dimensions plus grandes.

S

  d be the unit d-sphere of R d+1 endowed with the induced metric ∥ • ∥ S from R d+1 . A d-spherical cap α of center γ ∈ S d and spherical radius ρ ∈ (0, 2π) is a subset given by α = {x ∈ S d | ∥x -γ∥ S ≤ ρ} (1.1) which gives a partition of S d in three disjoint subsets: the interior of α, points of S d satisfying (1.1) strictly, the exterior of α, points of S d not satisfying (1.1) and the boundary of α, ∂α, points of S d satisfying the equality of (1.1). Let Caps(S d ) denote the family of d-spherical caps. It is well known that S d is homeomorphic to R d under the stereographic projection, where R d := R d ∪ {∞} is the one-point compactification of R d . We call a d-ball of R d the image of a d-spherical cap under the stereographic projection. We denote by Balls( R d ) the space of d-balls, isomorphic to Caps(S d ) given by the above construction. A d-ball b is called solid sphere, hollow sphere or half-space depending on whether the pole of the stereographic projection lies in either the exterior, the interior or the boundary of the corresponding d-spherical cap α b . Therefore, we have that a d-ball of R d of curvature κ ∈ R corresponds to one of the following subsets:

1. 1 . The space of oriented hyperspheres 7 1. 1 . 1 . 1 Figure 1 . 1 )Figure 1 . 1 :-

 171111111 Figure 1.1). The intersection angle does not depend on the choice of the point in the intersection.

Let H d+1 be( 1 . 2 )

 12 the Poincaré ball model of the hyperbolic space of dimension d + 1 embedded in R d+1 as the standard unit (d + 1)-ball. The boundary ∂H d+1 is exactly the unit sphere S d . A d-hyperbolic half-space of H d+1 is the intersection h := H d+1 ∩ b h where b h is a (d + 1)-ball orthogonal to H d+1 . We denote by Halfs(H d+1 ) the space of hyperbolic half-spaces of H d+1 . At the boundary of H d+1 , the intersection ∂H d+1 ∩ b h = S d ∩ b h is a d-spherical cap α h which corresponds to a d-ball b h by the stereographic projection. For any d-ball, the mapping h → α h → b h can be reversed so we can define the following isomorphisms: Balls( R d ) Caps(S d ) Halfs(H d+1 ) ≃ ≃ The notions of interior, exterior and boundary are easily extended for d-hyperbolic half-spaces. For d > 1, two d-balls b and b ′ have intersecting boundaries if and only if the corresponding d-hyperbolic half-spaces h b and h b ′ have intersecting boundaries. Moreover, the intersection angle of b and b ′ is equal to the dihedral angle of h b and h b ′

, 1 then 1 . 1 .

 111 Gram(B) is the matrix of the Lorentzian product in the basis B. The Lorentzian product of two vectors x, y ∈ L d+1,1 can be computed by ⟨x, y⟩ = c B (x) T Gram(B)c B (y) (1.3) The space of oriented hyperspheres where c B (x) is the column matrix of the coordinates of x in the basis B. By a fundamental theorem of bilinear forms (see [18, Theorem 2.1]), there is always an orthonormal basis B 0 = {e 1 , . . . , e d+2 } of L d+1,1 such that Gram(B 0 ) = diag(1, . . . , 1, -1).

Figure 1 . 2 :

 12 Figure 1.2: Geometric interpretation of the isomorphisms between the different models of the space of d-balls.

1 . 1 .

 11 if b and b ′ are disjoint where h b and h b ′ are the corresponding hyperbolic half-spaces and d H (h b , h b ′ ) is the hyperbolic distance between ∂h b and ∂h b ′ . In particular, if the Lorentzian vectors of b and b ′ are not both past-directed, then The space of oriented hyperspheres 11 ⟨b, b ′

  if b and b ′ are disjoint = -1 if b and b ′ are externally tangent = 0 if b and b ′ are orthogonal = 1 if b and b ′ are internally tangent > 1 if b and b ′ are nested (1.5)

  morphisms of R d mapping d-balls to d-balls [101]. An element of the Möbius Group is called a Möbius transformation. For every solid or hollow (resp. half-space) d-ball b we define the inversion on b, denoted by s b , as the sphere inversion (resp. Euclidean reflection) on the boundary of b. Alternatively, s b can be defined as the only Möbius transformation that maps b to -b and fixes a d-ball b ′ if and only if b ′ is orthogonal to b [101]. It is well-known that Möb( R d ) is generated by the set of inversions. The product s b s b ′ of the inversions on two d-balls centered at the origin with non-zero curvatures κ and κ ′ gives a rescaling of R d with scaling factor (κ ′ /κ) 2 . Thus, the group of Euclidean isometries and rescalings of R d is a subgroup of Möb( R d ).

  Figure 1.3). This embedding is usually called the carrier of the d-ball packing. The

Figure 1 . 3 :

 13 Figure 1.3: A circle packing with its carrier.

AFigure 1 . 4 :

 14 Figure 1.4: Example of a standard transformation.

Figure 1 . 5 :

 15 Figure 1.5: (Left) A knot diagram of the trefoil (denoted by 3 1 in the Alexander-Briggs notation): the simplest non-trivial knot, (right) a link diagram of the Hopf link (denoted by 2 2 1 ) : the simplest non-trivial link.

Figure 1 . 6 : 1 Figure 1 . 7 :

 16117 Figure 1.6: A 2-tangle diagram.

Figure 1 . 8 :

 18 Figure 1.8: Sum of tangles.

Figure 1 . 9 : 1 . 2 .

 1912 Figure 1.9: Unary operations of tangles.

Figure 1 . 10 :

 110 Figure 1.10: The tangle closures.

Figure 1 . 11 :

 111 Figure 1.11: The rational tangle t(2, -2, -3) obtained by Conway's algorithm.

Chapter 1 .

 1 Figure 1.13: A 3-braid (left), its closure (center) and the corresponding squaregrid diagram (right).

Figure 1 .

 1 Figure 1.14 an arbitrary polygonal, lattice and necklace representation of the trefoil knot.

Figure 1 .

 1 Figure 1.14: A polygonal representation (left), a lattice representation (center) and a necklace representation (right) of the trefoil knot.

Figure 1 . 15 :

 115 Figure 1.15: From left to right: a minimal polygonal representation with 6 segments, a minimal lattice representation with 12 segments [32] and a conjectured minimal necklace representation with 12 spheres [75] of the trefoil knot.

Figure 2 . 1 :

 21 Figure 2.1: From left to right: a diagram of the Figure-eight knot, the graph obtained from the diagram, same graph drawn with its medial graph and the patchwork graph of the initial knot diagram.

Proposition 2 . 3 . 1 .

 231 M ⊠ is a family of one real parameter.Proof. Let [D ⊠ ] -1 x (κ 1 ) = {d x , d 1 , d 2 , d -1 , d -2 }be a standard square-pyramidal circle packing where d 2 and d -2 are two unit disks tangent to the half-spaces d -1 = {y ≥ 1}, d x = {y ≤ -1} and d 1 is a disk of curvature κ 1 ∈ R tangent to d 2 , d -2 and d x .

Figure 2 . 3 :

 23 Figure 2.3: Extreme cases with an extra edge (top figures) and the equivalence classes of M 2 (⊠) (bottom figures).

2

 2

  .

4 .

 4 The equalities (a), (b) and (c) tell us that a square-pyramidal circle packing has essentially two different standard curvatures κ 1 and κ 2 which are inversely proportional and the smaller standard curvature must verify 1 < κ ≤ 2. We define the closest disjoint pair of D ⊠ as the disjoint pair {d i , d -i } satisfying that κ = κ i . The other disjoint pair will be called the farthest disjoint pair. In the following we use the indices {d c , d -c } and {d f , d -f } with {c, f } = {1, 2} and c ̸ = f to denote the closest and the farthest disjoint pair of D ⊠ . By convention, we define c = 1 and f

  as the arrangement of oriented spheres formed by: • The square-pyramid sphere packing B ⊠ obtained by inflating D ⊠ . • The mirror spheres b * 1 and b * 2 : the inflating of the mirror disks d * 1 and d * 2 respectively. • The tangency sphere b t : the inflating of the tangency disk d t . • The bridge spheres b ε3 and b ′ ε3 where: (i) b ε3 is the unique sphere externally tangent to b c , b f , b x , internally tangent to b * c and contained in the half-space {εz ≥ 0}, where {b c , b -c } and {b f , b -f } denotes the closest and the farthest pair of B ⊠ . (ii) b ′ ε3 is the sphere obtained by the inversion of b ε3 on the mirror-sphere b * c .

Figure 2 . 6 :Lemma 2 . 4 . 1 . 2 . 4 . The crossing sphere arrangement 35 2,

 262412435 Figure 2.6: The corresponding crossing sphere arrangement of squarepyramidal disk arrangements of Figure 2.5.

  the corresponding sphere b ∈ B P (L) \ B ⊠ must be disjoint to b t and thus, b is disjoint to X . Since, by Lemma 2.4.1, b εx3 , b ′ εx3 ⊂ X , we have that b εx3 and b ′ εx3 are disjoint to every sphere of B P (L) \ B ⊠ . On the other hand, Lemma 2.4.1 ensures that b εx3 and b ′ εx3 are at most tangent to every sphere of B ⊠ . Claim (3)] We first notice that, by Lemma 2.4.1, the bridge spheres of a crossing sphere arrangement are externally tangent. It remains to show that the bridge spheres of different crossing sphere arrangements do not overlap. Let x and x ′ be two different crossing vertices of P (L) and let D

2 = 8 1

 28 These values are given in the proof of Lemma 2.4.1 in terms of the smaller standard curvature by λ -c,3 = -1-2 √ κ and λ z,3 = 3 + 2 √ κ -κ. The smaller standard curvature can be computed by using Lemma 2.3.2 (a) obtaining κ = 1-λ c,-c -λ f,-f where λ c,-c := ⟨b c , b -c ⟩ and λ f,-f := ⟨b f , b -f ⟩. We use the well-known algorithm of Collins and Stephenson [25] for the construction of circle packings from the tangency graph. The radius of the outer disks and the visual precision can be chosen. The necklaces representations of the Figures 2.10 and 2.11 are obtained by this algorithm with outer radii equal to 1 and precision 10 -4 .

  If the thread made by the bridge sphere is under-crossing in L, then change the sign of the third coordinate of b 3 (x) and b

Figure 2 . 11 : 43 3. 2 45 3. 3 46 3. 4 49 3. 4 . 1 50 3. 5 51 3. 6 52 3. 6 . 1 57 3. 6 . 2 58 3. 7 60 3. 7 . 1

 2114324534644941505516526157625876071 Figure 2.11: Necklace representation of the knot 8 17 with 40 spheres.

Figure 3 . 1 :

 31 Figure 3.1: From left to right: the classic Apollonian circle packing, the Guettler-Mallows' generalization and the Apollonian ring packing with n = 5.

3. 2 . 45 3. 2

 2452 The projective model of the space of oriented hyperspheres. The projective model of the space of oriented hyperspheres.

E

  d+1 will be called an outer-sphere point. The reciprocal bijection between an outersphere point u ∈ E d+1 and a d-ball b(u) ∈ Balls( R d ) can be obtained geometrically by taking a light source which illuminates S d from u. The spherical illuminated region is a spherical cap which becomes, under the stereographic projection from the North Pole, the d-ball b(u). We say that u is the light source of b(u) and, reciprocally, we say that b(u) is the illuminated region of u.

Figure 3

 3 Figure 3.2: A light-source and its illuminated region in the projective model.

  {p 1 , p 2 , . . . , p d } is the Schläfli symbol of P. The symmetric group of P is then the spherical Coxeter group with Coxeter graph p 1 p d . All polygons admit a regular realization. The Platonic solids, namely, the tetrahedron T 3 , the octahedron O 3 , the cube C 3 , the icosahedron I 3 and the dodecahedron D 3 are the five regular 3-polytopes (polyhedra). The 4-simplex T 4 , the orthoplex O 4 , the hypercube C 4 , the 600-cell I 4 and the 120-cell D 4 are the five regular 4-polytopes which can be thought as a 4-dimensional analogue of the Platonic solids. The remaining regular 4-polytope, the 24-cell R 4 (the notation is not standard), completes the list of regular 4-polytopes. For every d ≥ 2, we shall denote by T d+1 , O d+1 and C d+1 the (d + 1)-dimensional analogues of the tetrahedron, octahedron and cube. These are usually called (d + 1)-simplex, (d + 1)-cross-polytope and (d + 1)-cube, respectively. It is well-known that in dimension 5 or above, these three families are the only regular polytopes [29].

3. 4 . 49 3. 4

 4494 The ball-arrangement projection. The ball-arrangement projection.

  d-balls whose light sources are the vertices of P. Reciprocally, we say that P is the light-source polytope of the ball arrangement β(P). Chen proved in [21, Th. 5.2] that if β(P) is a packing, then the tangency graph of β(P) is a spanning subgraph of the graph of P.

Figure 3 . 3 :

 33 Figure 3.3: (Left) An outer-sphere polyhedron P with the spherical illuminated regions of its vertices; (right) the ball-arrangement projection β(P).

Figure 3 . 4 :

 34 Figure 3.4: (Left) A circle packing which is not polytopal; (center) its lightsource polyhedron; (right) same polyhedron with the unit sphere. The edge cutting the sphere does not correspond to any edge of the tangency graph.

For every d ≥ 1

 1 and 0 ≤ i ≤ d, we define a Centered Ball Packing (i-CBP) projection of a regular (d + 1)-polytope P, as the ball-arrangement projection of an edgescribed regular realization of P containing an i-face whose barycenter is in the ray going from the origin to the North Pole of S d (∞ in R d ). We show in the Appendices (Tables A.2 and A.3) the CBP projections of the Platonic solids and the regular 4-polytopes.

3. 5 . 51 3. 5

 5515 Duality of polytopal d-ball packings Duality of polytopal d-ball packingsFor d ≥ 2, let P be an edge-scribed (d + 1)-polytope. We can always send P by a projective transformation preserving the sphere into an edge-scribed (d + 1)polytope P 0 containing the origin in its interior. Therefore, for any polytopal d-ball packing B P there is a Möbius transformation µ such that µ(B P ) = β(P 0 ). We define the dual of B P as the ball arrangement B * P := µ -1 (β(P * 0 )). The dual B * P does not depend on the choice of P 0 . For any vertex v of P 0 , and for any vertex of P * 0 corresponding to a facet f of P 0 containing v, the corresponding d-balls b v ∈ B P and b f ∈ B * P are orthogonal (this can be easily obtained by combining Equations (1.5), (3.3) and (3.5)). Since P is edge-scribed, P 0 is edge-scribed and P * 0 is ridgescribed. In the case d = 2, P * 0 is also edge-scribed and therefore B P and B * P are both circle packings. The union B P ∪ B * P has been called a primal-dual circle representation of P[START_REF] Felsner | On primal-dual circle representations[END_REF]. We show in Figure3.5 a primal-dual circle representation of the icosahedron. The Midsphere Theorem of Brightwell and Scheinerman[START_REF] Brightwell | Representations of planar graphs[END_REF] gives the existence and the uniqueness up to Möbius transformations of primal-dual circle representations for every polyhedron. It can be thought of as a stronger version of the Koebe-Andreev-Thurston Circle packing theorem[START_REF] Bobenko | Variational principles for circle patterns and Koebe's theorem[END_REF].

Figure 3 .

 3 Figure 3.5: (Left) An edge-scribed icosahedron I and its polar I * ; (center) the spherical illuminated regions of I and I * ; (right) a primal-dual circle representation of I.

Proposition 3 . 6 . 1 .

 361 Let B and B ′ be two d-ball packings with maximal rank. Then B is Möbius equivalent to B ′ if and only if Gram(B) = Gram(B ′ ). Proof. The necessity is trivial since Möbius transformations preserve the inversive product. Now let us suppose that Gram(B) = Gram(B ′ ) is a matrix of rank d + 2.

Remark 4 .Corollary 3 . 6 . 1 .Chapter 3 .

 43613 Therefore, µµ 0 induces a Möbius transformation mapping B to B ′ . Proposition 3.6.1 does not hold for arrangements of d-balls in general. To see this, one may consider any ball packing B = {b 1 , . . . , b n } whose Gramian is maximal. The arrangement -B := {-b 1 , . . . , -b n } is not a packing, and therefore B and -B are not Möbius equivalent. However, Gram(B) = Gram(-B). In practice, we shall use Lemma 1.1.1 and Proposition 3.6.1 to study Möbius uniqueness. As an example, the result of Wilker [104, Lemma 6] stating that any two d-ball packings made by d + 2 pairwise tangent d-balls are Möbius equivalent, follows directly from Proposition 3.6.1. In terms of the tangency polytope, this result can be restated as follows. For every d ≥ 1, the (d + 1)-simplex is Möbius unique. It is clear that all polygons are edge-scribable but not all are Möbius unique. Corollary 3.6.2. The only polygon which is Möbius unique is the triangle. Polytopal ball packings Proof. Let P be an edge-scribed n-gon with n > 3 and let [B P ] n 1 = (b 1 , b 2 , . . . , b n ) be the standard 1-ball packing obtained by applying a standard transformation to the ball-arrangement projection of P. Let -1 < x < 1 be the contact point of the two closed intervals b 2 and b 3 . By replacing b 2 and b 3 by two closed intervals b ′ 2 and b ′ 3 obtained by moving x a suitable distance we can construct another 1-ball packing [B ′ P ] n 1 which is not Euclidean congruent to [B P ] n 1 . By Lemma 1.1.1, [B P ] n 1 and [B ′ P ] n 1

Theorem 3 . 6 . 1 .

 361 Let d ≥ 2 and let B P be a polytopal d-ball packing where P is an edgescribed realization of the (d + 1)-cross polytope or (d + 1)-cube. Then, for every two vertices u, v of P, we have

Proof. We proceed by induction on d ≥ 2 .

 2 The initial case d = 2 can be easily checked in a single octahedral or cubical circle packing. Since, by Cor. 3.6.3, polyhedra are Möbius unique, then Eq. (3.7) holds for every edge-scribed realization of the octahedron or cube. Let us now suppose that (3.7) holds for any edge-scribed realization of the (d+1)cross polytope or (d + 1)-cube for some d ≥ 2.

3. 6 . 55 (

 655 Möbius uniqueness of edge-scribable polytopes P = O d+2 ) Let B O d+2 be a polytopal (d + 1)-ball packing where O d+2 ⊂ E d+2 is an edge-scribed realization of the (d + 2)-octahedron. We give to the vertices of O d+2

  we consider the following collection of (d + 1)-ballsB i,j := {b 1 , . . . , b d+1 , b -i , b -j } ⊂ B O d+2 where b k := b v k . Since B O d+2 is polytopal, the tangency graph of B O d+2 is the 1skeleton of O d+2 .Therefore, b d+2 and b -(d+2) are tangent to every b k ∈ B i,j . By (1.5), we have that b d+2 and b -(d+2) satisfy ⟨b k , b⟩ = -1 for every b k ∈ B i,j (3.8) and b ∈ {b d+2 , b -(d+2) }. In inversive coordinates, (3.8) becomes the following linear system

2 .

 2 perplane H containing all the vertices V (O d+2 ) \ {v d+2 , v -(d+2) }. We thus can find a Möbius tranformation µ ∈ Möb(S d+1 ) such that the corresponding projective transformation sends H to the hyperplane {x d+2 = 0} ⊂ E d+2 . After identifying µ(H) with E d+1 we obtain that µ(H ∩ O d+2 ) becomes an edge-scribed realization of the (d + 1)-octahedron O d+1 . The identification µ(H) ≃ E d+1 preserves the inversive product of the (d + 1)-balls corresponding to the points lying in H. Moreover, the distance between u and v in the graph of O d+2 is equal to the distance in the graph of O d+1 . By the invariance of the inversive product under Möbius transformations and the induction hypothesis, we have that Equation (3.7) holds for any two (d + 1)-balls of B O d+2 \ {b d+2 , b -(d+2) }. The same arguments work if we exchange b 1 and b d+2 in B i,j , so (3.7) holds in the remaining cases of B O d+2 .(P = C d+2 ) Let B C d+2 be a polytopal (d+1)-ball packing where C d+2 ⊂ E d+2 is an edgescribed realization of the (d + 2)-cube. Let u, v be two vertices of C d+2 . We notice that0 ≤ d G (u, v) ≤ d + 2. Let ū the unique vertex in C d+1 such that d G (u, ū) = d + 2.If v ̸ = u then there is a facet f of C d+2 containing u and v. Let H be the hyperplane obtained by the affine hull of f . As above, there is µ ∈ Möb(S d+1 ) such that the corresponding projective transformation sends H to the hyperplane {x d+2 = 0} ⊂ E d+2 , and therefore µ(H ∩ C d+2 ) induces an edge-scribed realization of the (d + 1)cube. By applying the induction hypothesis we obtain that Equation (3.7) holds for any b u , b v ∈ B C d+2 corresponding to two vertices of C d+2 at distance strictly less than d + It remains thus to show that (3.7) holds for b u and b ū. Let v 1 , . . . , v d+2 be the neighbours of u in the graph of C d+2 . We consider the (d + 1)-ball arrangement ∆ = (b u , b v 1 , . . . , b v d+2 , b ū) ⊂ B C d+2 . By the induction hypothesis, we have

3. 6 .Corollary 3 . 6 . 4 .Corollary 3 . 6 . 5 .

 6364365 Möbius uniqueness of edge-scribable polytopes 57 Since b u and b ū are disjoint then, by Equation (1.5), λ < -1 and therefore we have that ⟨b u , b ū⟩ = -3 -2d = 1 -2(d + 2) = 1 -2d G (u, ū). For every d ≥ 2, the (d+1)-cross polytope and the (d+1)-cube are Möbius unique. Proof. Let B P and B P ′ be two polytopal d-ball packings where P and P ′ are two edge-scribed realizations of the (d + 1)-cross polytope (resp. (d + 1)-cube). By Theorem 3.6.1, we can find an ordering such that Gram(B P ) = Gram(B P ′ ). The Möbius uniqueness then follows from Proposition 3.6.1. The 24-cell is Möbius unique. Proof. It is well-known that the 1-skeleton of a 24-cell admits a 3-coloring such that the vertices of each color span an orthoplex. By taking the vertices of two out of the three colors, we span a hypercube. Moreover, the edges of this hypercube are also edges of the initial 24-cell. Therefore, every polytopal sphere packing whose tangency polytope is the 24-cell contains a hypercubical sphere packing. Let B R 4 be the 0-CBP projection of the 24-cell. The even layers of B R 4 gives a 3-CBP projection B C 4 of the hypercube (see Table A.3 in the Appendices). Let B P be a polytopal sphere packing where P is another edge-scribed realization of the 24-cell and let Q ⊂ P be one of the hypercubes contained in P. Since, by Corollary 3.6.4, the hypercube is Möbius unique, then there is a Möbius transformation µ sending the packing B Q to B C 4 . Furthermore, every sphere b ∈ B R 4 \ B C 4 must be tangent to 6 spheres of B C 4 corresponding to the vertices of a facet of C 4 . This condition forces µ to send the spheres in B P \ B Q to B R 4 \ B C 4 implying that B P and B R 4 are Möbius equivalent.

Lemma 3 . 6 . 1 . 5 Figure 3 . 6 :

 361536 Figure 3.6: The sphere packing obtained by inverting the ball-arrangement projection of a glueing of three 4-simplices.

Chen mentioned in [ 21 ,Proposition 3 . 6 . 2 .

 21362 Section 5] similar arguments as above, by considering a 2-CBP projection of the 4-simplex instead of a 1-CBP projection. A natural generalization of stacked polytopes are the stacked P-polytopes, introduced by Chen and Padrol in [24], as polytopes obtained by connected sums of several copies of a given polytope P. By combining Corollaries 3.6.4 and 3.6.5 with Lemma 1.1.1, we can generalize the construction of Eppstein, Kuperberg and Ziegler by applying the same arguments to the stacked P-polytopes, where P = O 4 , C 4 , R 4 . The following 4-polytopes are not edge-scribable:

3. 7 three

 7 Apollonian packings with a minimal coloration. In the first case each orbit has a different color. If P is regular, then Sym(B P ) acts transitively on the d-balls of the Apollonian clusters Ω(B P ) and Ω(B * P ). Moreover, if S is a set of generators of Sym(B P ), then Equation (3.10) implies that for any s f ∈ S(B P ) and any s v ∈ S(B * P ) the group SA(B P ) is generated by S ∪ {s f } and SSA(B P ) is generated by S ∪ {s v , s f }.

Figure 3 . 7 :

 37 Figure 3.7: A tetrahedral (left), octahedral (center) and cubical (right) Apollonian packing with a minimal coloration.

Figure 3 . 9 :

 39 Figure 3.9: The circle packings B {p,3} for p = 3 (left), p = 4 (center) and p = 5 (right). In dashed line the generators of Sym(B {p,3} ) .

Theorem 4 . 1 . 2 (

 412 Soddy-Gosset). The curvatures of d + 2 pairwise tangent d-balls in R d satisfy

Theorem 4 . 2 . 1 (

 421 Polytopal Descartes' Theorem). Let B P be a polytopal d-ball packing where P is a regular edge-scribed (d+1)-polytope with d ≥ 1. Then, for any flag (v, e, . . . , f, P), the polytopal curvatures κ v , κ e . . . , κ f , κ P of B P satisfy Φ P (κ v , κ e , . . . , κ f , κ P ) = 0 (4.14)

  be a flag of P and let ∆ = (x f 0 , . . . , x f d , x P ) ⊂ L d+1,1 . We shall show that for any d ≥ 1, Gram(∆) = C(-L P (0), -L P (1), . . . , -L P (d + 1))(4.18)

1 ,

 1 then (i) and (ii) cover all the possible values for j. Let d > 1. First, we shall prove that (4.19) holds for j = d+1. Since P is edge-scribed, the facet f d intersects the unit sphere S d ⊂ E d+1 in a (d -1)-sphere S. Let µ ∈ Möb( R d ) given by the composition of the following two Möbius transformations: -A rotation of S d which sends the barycenter of f d to the line spanned by the North pole of S d . -The Möbius transformation corresponding to a rescaling of R d , which sends the rotated S to the equator of S d . By identifying the hyperplane {x d+1 = 0} ⊂ E d+1 with E d we have that µ(f d ) becomes a regular edge-scribed realization f d of the regular d-polytope f d . We can then apply again Lemma 4.2.3 to obtain

ering this time f d- 1

 1 as a facet of the regular edge-scribed d-polytope f d . By carrying on these arguments for the rest of the values 3 ≤ j ≤ d + 2 we obtain (4.19) which gives (4.18).

Corollary 4 . 2 . 2 (

 422 Soddy-Gosset). Let κ 1 , . . . , κ d+2 be the curvatures of a polytopal d-ball packing B T d+1 . Then,

For d = 2 , 3 , 1 . 4 . 2 . 4 .

 231424 2. The vertices corresponding to the curvatures are the vertices of a facet of O d+1 , which is a d-simplex. Therefore, we can find a flag (f 0 , . . . , f d , O d+1 ) where Eq. (4.26) is satisfied for every i = 0, . . . , d. The Polytopal Descartes' Theorem combined with the hyperoctahedral quadratic form gives the result. Eq. (4.27) is equivalent to the second equation given in the Guettler-Mallows and Dias-Nakamura generalizations (Theorems 4.1.3 and 4.1.4), respectively, where κ µ = κ O d+1 . The first equation in both theorems corresponds to the Antipodal relation of Lemma 4.2.Corollary For every d ≥ 1, let κ 1 , . . . , κ d+2 be the curvatures of d + 2 consecutive tangent d-balls of a polytopal d-ball packing B C d+1 where κ 1 and κ d+2 are the curvatures of two d-balls at distance d + 1 in the tangency graph of B C d+1 . Then we have d d+1 i=1

32 )

 32 Proof. The three equations are obtained by adding the numbers inside the parenthesis in (4.25), (4.27) and (4.28), respectively.

2

 2 + 5 2 + 52 2 ) = 74 2

Figure 4 . 1 :

 41 Figure 4.1: Three primitive solutions to Eq. (4.30), (4.31), (4.32) for d = 2, obtained by the relations between the polytopal curvatures given in Corollaries 4.2.2, 4.2.3 and 4.2.4, respectively.

Figure 4 . 2 :Corollary 4 . 3 . 1 .

 42431 Figure 4.2: Two integral tetrahedral Apollonian packings.

Corollary 4 . 3 . 2 .

 432 The set of curvatures of a Platonic Apollonian packing Ω(B P ) can be obtained from the curvatures of three consecutive tangent disks of B P corresponding to three vertices of P lying in the same face.

Corollary 4 . 3 . 3 (

 433 Integrality condition for octahedral Apollonian packings). Let κ 1 , κ 2 , κ 3 be the curvatures of the three pairwise tangent disks of an octahedral circle packing B O 3 .

Corollary 4 . 3 . 4 .

 434 Cubical Apollonian packings have been studied by Stange in [95] as a particular case of the Schmidt arrangement Q[ √ -2]. They are also mentioned at the end of the paper of Apollonian ring packings of Bolt et al. [10]. Before giving the corresponding integrality condition, let us discuss some useful relations between the curvatures of a cubical circle packing. Since the cube is centrally symmetric, the Antipodal relation of Lemma 4.2.1 holds. Moreover, by applying the Antipodal relation to any square-face we obtain the following relation (which was already noticed by Stange in [95]). For any cyclic chain of tangent disks d 1 , d 2 , d 3 , d 4 of a cubical circle packing, the curvatures of the disks satisfy

Corollary 4 . 3 . 5 (

 435 Integrality condition for cubical Apollonian packings). Let κ 1 , κ 2 , κ 3 be the curvatures of three consecutive tangent disks in a cubical circle packing B C 3 .

Figure 4 . 4 :

 44 Figure 4.4: Two cubical Apollonian packings with initial curvatures (-1, 2, 3) and (-3, 5, 12).

4. 3 .

 3 Integrality of the Platonic Apollonian packings[START_REF] Rasskin | Polytopack: a framework for polytopal sphere packings[END_REF]

Figure 4 . 5 :

 45 Figure 4.5: Two icosahedral Apollonian packings with initial curvatures (-1, φ, 2φ) (left) and (0, 0, 1) (right).

Corollary 4 . 3 . 7 .

 437 For any cyclic chain of tangent disks d 1 , d 2 , d 3 , d 4 , d 5 of a dodecahedral circle packing, their curvatures satisfy

Figure 4 . 6 : 87 5. 2 88 5. 2 . 1 90 5. 2 . 2 91 5. 3 94 5. 3 . 1

 46872882190229139431 Figure 4.6: Two φ-integral dodecahedral Apollonian packings with initial curvatures (-1, φ + 1, 2φ) (left) and (0, 0, 1) (right).

Figure 5 . 1 :

 51 Figure 5.1: The orthoplicial Apollonian sphere packing at 0, 1 and 2 iterations. Each color represents an orbit.

Figure 5 . 2 : 2 . 5 .

 5225 Figure 5.2: The standard orthoplicial sphere packing [B O 4 ] 1 2 .

Figure 5 . 3 :

 53 Figure 5.3: An orthoplicial trinity containing the standard [B O 4 ] 1 2 .

Figure 5 . 4 :

 54 Figure 5.4: From left to right: a 3-CBP projection, a 0-CBP projection and another 3-CBP projection of the orthoplex, and the orthoplicial trinity formed by the three.

2 )Corollary 5 . 2 . 1 .

 2521 is generated by {r v , r e , r r , r f , s f }, where s f is the inversion on the sphere orthogonal to b 1 , b 2 , b 3 and b 4 . By using the inversive coordinates, we obtain a linear faithful representation of SA([B O 4 ] 1 2 ) as the discrete subgroup of O ↑ 4,1 (Q), where the generators {r v , r e , r r , r f , s f } are represented by the matrices {V, E, R, F, S}, respectively. The relations in the Coxeter graph can be checked by straightforward computations on the matrices. By the Möbius uniqueness of the orthoplex, the symmetrized Apollonian group of any other orthoplicial ball packing is isomorphic to SA([B O 4 ] 1 2 ). The orthoplicial Apollonian group is isomorphic to the subgroup of O ↑ 4,1 (Z) generated by the following 16 matrices:

Proposition 5 . 3 . 1 .

 531 two different polytopal Apollonian clusters Ω(B P ) and Ω(B P ′ ), respectively, are said to be algebraically equivalent if Γ and Γ ′ are isomorphic and there is an equivariant bijection between Γ • X and Γ ′ • X ′ with respect to the actions. More specifically, Σ and Σ ′ are algebraically equivalent if there exist two bijections ϕ : Γ → Γ ′ and ψ: Σ → Σ ′ such that (i) ϕ : Γ → Γ ′ is a group isomorphism (ii) For all g ∈ Γ and all b ∈ X, ψ(g • b) = ϕ(g) • ψ(b)If in addition ψ preserves curvatures, we say that Σ and Σ ′ are arithmetically equivalent. We shall say that an Apollonian cluster has a tetrahedral, octahedral, cubical, icosahedral or dodecahedral Apollonian section if it contains an Apollonian section algebraically equivalent to an Apollonian packing of the corresponding Platonic solid. Every orthoplicial Apollonian packing contains a tetrahedral, octahedral and cubical Apollonian section.Proof. Since the orthoplex is Möbius unique, it is enough to find the desired Apollonian sections in a particular orthoplicial Apollonian packing. First, we shall construct a tetrahedral and octahedral Apollonian section in Ω([B O 4 ] 12 ) (Fig.5.5).

Figure 5 . 5 :

 55 Figure 5.5: The orthoplicial Apollonian packing Ω([B O 4 ] 12 ) at depth ≤ 3, view from above.

(

  Octahedral section) We may apply a similar strategy to construct an octahedral section in Ω([B O 4 ] 1 2 ) by intersecting with the plane H O := {x -y = 1}, orthogonal to every b ∈ X O := [B O 4 ] 1 2 \ {b 4 , b -4 }. The intersection H O ∩ X O gives an octahedral circle packing B O 3 . We define then Σ

3 )

 3 is given by t ijk → s ijk , where s ijk denotes the inversion on the disk orthogonal to the three disks{b i ∩ H O , b j ∩ H O , b k ∩ H O }. The equivariant bijection ψ O : Σ O → Ω(B O 3) is then given by g(b) → g(b) ∩ H O . We illustrate in Figure 5.6 the tetrahedral and octahedral sections Σ T and Σ O .

Figure 5 . 6 :

 56 Figure 5.6: A tetrahedral (left) and octahedral (right) Apollonian section of Ω([B O 4 ] 1 2 ).

(

  Cubical section) This case is a bit more tricky. We consider the orthoplicial sphere packing B ↑x O 4 obtained by a 3-CBP projection of the orthoplex, with the labelling and coordinates given in Figure5.7. We call the labelling given in Figure5.7 the x-labelling. We notice that in the x-labelling, i is positive if and only if the first coordinate of the center of b i is positive.

Figure 5 . 7 : 3 1 √ 2 ( 1 , 1 , 1 ,, the 98 Chapter 5 .

 57312111985 Figure 5.7: The orthoplicial sphere packing B ↑x O 4 .

Figure 5 . 8 :

 58 Figure 5.8: (Left) B ↑x O 4 in gray with B ⊥ H C in blue; (right) B C 3 in gray with B * C 3 in blue.

  C 3 ) given above with a 2-coloring. Then, two disks ψ C (b) and ψ C (b ′ ) have the same color if and only if the centers of b and b ′ both lie on the same side of the plane H C . When two disks with the same color (and therefore disjoint), correspond to two vertices lying in the same square-face, then the corresponding spheres are tangent (see Fig. 5.9).

Figure 5 . 9 :

 59 Figure 5.9: (From left to right) The orthoplicial Apollonian sphere packing Ω(B ↑x O 4 ); a cubical Apollonian section Σ C of Ω(B ↑x O 4 ); Σ C zoomed; the corresponding cubical Apollonian circle packing with a minimal coloration.

4 I 4 M 3 )

 443 -1 i(b) where I 4 is the identity matrix of size 4 = k 5 0 i(b) = k 4 i(b) = κ(b) and Σ ′ T ⊂ Ω(B ′ O 4 ) are arithmetically equivalent. We may use the beginning of the proof to show the octahedral case in a similar way as above. In this case, we may consider the orthoplicial sphere packing obtained from [B O 4 ] 1 2 after a rotation around the z-axis of angle π 4 , so the plane H O becomes the plane {x = 0}. Then, the bijection ψ -1 O : Ω(B O 3 ) → Σ O has the same expression in inversive coordinates as ψ -1 T and the rest of computations are identical as in the tetrahedral case. As before, the cubical case is more delicate. Let B ↑x O 4 be the orthoplicial sphere packing of Figure 5.7 and let Σ C = Γ C • B ↑x O 4 be the cubical Apollonian section described in the proof of Prop. 5.3.1. Let B C 3 be the cubical circle packing given by ψ C (B ↑x O 4 ). We define the mapping ε : B C 3 -→ {1, -1} b -→ sign(x 1 (b)) (5.2) where x 1 (b) is the first coordinate of the center of ψ -1 C (b). Then, in inversive coordinates, the bijection ψ -1 C : B C 3 → B ↑x O 4 and the group isomorphism ϕ -1 C : A(B C 3 ) → Γ C are given by

Chapter 5 .

 5 Apollonian sections of the orthoplicial Apollonian packing definition. Then, for any b ∈ Ω(B ′ C 3 ) we have

(

  Cubical section) Let κ 1 , κ 2 , κ 3 , κ 4 be the curvatures of four consecutive tangent disks b 1 , b 2 , b 3 , b 4 of B ′ C 3 with b 4 tangent to b 1 . By Cor. 4.3.4, we have

Figure 6 . 1 :

 61 Figure 6.1: Two necklace representations of the Hopf link: the Soddy's Hexlet (left) and the orthoplicial sphere packing (right).

Figure 6 . 2 :

 62 Figure 6.2: (Left) [B O 4 ] 1 2 with the generators of Σ 12 , view from above; (top right) a square-grid braid diagram of the trefoil knot and the corresponding embedding of the trefoil in the carrier of Σ 12 (bottom right).

Corollary 6 . 2 . 1 .Chapter 6 .

 6216 Let L be a link with braid index equal to 2. Then, ball(L) ≤ 4cr(L) 108 Orthoplicial Apollonian packings and rational links Proof. Let L be a link equivalent to a closed 2-braid with n crossings. The corresponding square-grid diagram is a reduced alternating diagram of L. Therefore, by Th. 1.2.1, cr(L) = n. On the other hand, by using the half-spaces b 1 and b 2 in Σ 12 for closing the 2-braid, we can obtain a necklace representation of L with 4n spheres.

  5.3.1, we consider the orthoplicial sphere packing B ↑z O 4 obtained by relabelling of B ↑x O 4 according to the Figure 6.3. We call the given labelling the z-labelling. Analogously to the x-labelling, a label i is positive in the z-labelling if and only if the third coordinate of the center of b i is positive.

  b

Figure 6 . 3 : 4 . 6 . 3 .

 63463 Figure 6.3: The orthoplicial sphere packing B ↑z O 4 .

Figure 6 . 4 :Chapter 6 .

 646 Figure 6.4: (Top left) B ↑z O 4 with its carrier and the plane {z = 0}, view from above; (top right) the corresponding cubical circle packing B C 3 with the zcoloring; (bottom left) the tangency graph of B ↑z O 4 with the z-coloring; (bottom right) the tangency graph of B C 3 with the z-coloring.

Figure 6 . 5 :

 65 Figure 6.5: An orthocubical necklace representation of the trefoil knot (left) and the corresponding cubical diagram (right).

  6.6).

12 Figure 6 . 6 :

 1266 Figure 6.6: The disks of the dual B * C 3 in blue and the walls of the symmetries {r 12 , r 13 , r 23 , r -13 , r -23 , r -33 } in dashed lines.

Figure 6 . 10 :

 610 Figure 6.10: The orthocubical tangle closures.

we have the following. Proposition 6 . 3 . 1 .Theorem 6 . 3 . 1 .

 631631 Any algebraic link admits a necklace representation in the orthocubical sphere packing. The orthocubical version of Conway's algorithm also allows us to improve the upper bound on the ball number of rational links. Let L be a rational link. Then ball(L) ≤ 4cr(L)

  from the definition of the orthocubical tangle operations that| t ′ | = 4(a 1 + . . . + a n -1) + | t 1 | = 4(a 1 + . . . + a n ) = 4cr(L)by Eq. (1.20).

6. 4 . A geometric interpretation of continued fractions 117 Figure 6 . 11 :

 4117611 Figure 6.11: An orthocubical representation of the Pretzel knot P (3, -2, 3) with 28 spheres.

c 3 e 3 / 2 η 3 / 2 Figure 6 . 12 :

 33232612 Figure 6.12: The projection of the first edge (red) and first tangency point (green) of the rational tangle t 3/2 in a cubical diagram.

( 1 . 1 √ 2 )

 112 11) (we chose λ = and (1.14), which give i(η ∞ ) = Consequently, m k is represented by the matrix M(k) := (S 1 R 13 ) k R 12 . By induction on k, we find that

1 •••+ 1 an 2 (r -s) 2 √ 2 (r 2 -

 112222 theorem. Now let us suppose the theorem to be true for n-1. Let r/s = a 2 + 1 )M(a 2 ) • • • M(a n ) rs + s)

Corollary 6 . 4 . 1 . 2 ) 3 Figure 6 . 13 :

 64123613 Figure 6.13: The cubical Apollonian packing Ω(B C 3 ) under the z-coloring with plot range [0, 1 + √ 2] × [0, 2( √ 2 -1)], together with the first tangency point η 5/3

Figure 7 . 14 :

 714 Figure 7.14: An integral tetrahedral (left) and a φ-integral icosahedral (right) Apollonian packings both with initial curvatures (-4, 8, 9).

7. 2 . Integral packings 125 Question 3 .

 21253 What is the maximum number of distinct integers that one can find in an icosahedral Apollonian packing?

7. 16

 16

  , 7.17 and 7.23 the three constructions.

Figure 7 . 16 :

 716 Figure 7.16: The Hopf link Hf (2) (left), and the construction of a necklace representation of Hf (2) with 8 spheres contained in a 0-CBP projection of the orthoplex.

Figure 7 . 17 :

 717 Figure 7.17: The link Hf (4) (left) and the construction of a necklace representation of Hf (4) in a 0-CBP projection of the 24-cell.

7. 5 .Figure 7 . 18 :

 5718 Figure 7.18: The link Hf (12) (top left) and the construction of a necklace representation of Hf (12) in a 0-CBP projection of the 600-cell.

Figure 7 . 20 :

 720 Figure 7.20: Manipulation of a polytopal circle packing with Polytopack.

Figure 7 . 22 :

 722 Figure 7.22: Deformations of 3D-meshes with sphere packings preserving the tangency graph.

Figure 7 . 23 :

 723 Figure 7.23: Deformations of knots and links preserving the knot-type.

Figure 7 . 24 :

 724 Figure 7.24: Projective transformations of edge-scribed polyhedra preserving edge-scribedness.

Figure 7 . 25 :

 725 Figure 7.25: Projective transformations of inscribed polyhedra preserving inscribedness.

Figure 7 . 26 :

 726 Figure 7.26: Study of internal structures of the ball-arrangement projections of 4-polytopes.

1.1.7 Möbius uniqueness of

d-ball packable graphs We say that two d-ball packings are Möbius equivalent if one can be sent to the other by a Möbius transformation. If such a Möbius transformation is a Euclidean isometry then we say that the two packings are Euclidean congruent. A d-ball packable graph G will be Möbius unique if all the d-ball packings whose tangency graph is G are Möbius equivalent. A useful result to detect Möbius uniqueness is the following: Lemma 1.1.1. Let B G and B ′ G be two d-ball packings with same tangency graph G and let ij be an edge of G. Then B G and B ′ G are Möbius equivalent if and only if [B G ] i j and [B ′ G ] i j are Euclidean congruent.

Proof. Let ϕ : B

An upper bound on the ball number Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 From links to circle packable graphs . . . . . . . . . . . . . . . . . . . 28 2.3 Square-pyramidal disk arrangements . . . . . . . . . . . . . . . . . . . 29 2.4 The crossing sphere arrangement . . . . . . . . . . . . . . . . . . . . . 33 2.5 The proof of the upper bound . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 An algorithm for knotted necklaces . . . . . . . . . . . . . . . . . . . . 40 2.1 Introduction In this chapter, we present the following upper bound on the ball number. Theorem 2.1.1. For

	CHAPTER 2
	. In Chapters 2 and 6, we shall
	present different methods of construction of necklace representations. The first gives
	a linear upper bound on the ball number of nonsplittable and nontrivial links, in
	terms of the crossing number. The last method will improve this upper bound for
	rational links with the help of the polytopal sphere packings introduced in Chapter
	3.
	any nontrivial and nonsplittable link L, we have that
	ball(L) ≤ 5cr(L)
	Let us give a sketch of the proof. We start by considering a minimal crossing
	diagram of a link L. Then, we construct a simple planar graph with 3cr(L) vertices
	containing a subgraph admitting a planar embedding isotopically equivalent to a
	planar projection of L. By applying the KAT theorem, and then the inflate opera-
	tion, we shall construct a sphere packing whose carrier contains the previous planar
	projection of L. Finally, for each crossing of the planar projection, we reconstruct the
	projected crossing by adding two spheres that serve as a "bridge", and in this way,

Chapter 2. An upper bound on the ball number we

  obtain a necklace representation of L with 5cr(L) spheres.

which passes through the tangency point d x ∩ d 2 . The orientation of the interior is determined by the condition d 1 ⊂ d * 1 .

  The tangency disk d t : the disk whose boundary passes through all the tangency points of d 1 , d 2 , d -1 and d -2 , and satisfies d x ⊂ d t . , d -1 and d x must be a half-space with the y-axis as boundary. As before, the orientation of the interior comes from the condition d 2 ⊂ d * For d t , by symmetry, the only circle passing through the tangency points d 1 ∩ d 2 , d 1 ∩ d -2 and d -1 ∩ d 2 must passes through d -1 ∩ d -2 . Again, the orientation is determined by the condition d x ⊂ d t .

	* 1 , d * 2 , d t )
	formed by
	• D ⊠ = {d x , d 1 , d 2 , d -1 , d -2 }: a square-pyramidal circle packing.
	• The mirror disks d * 1 and d * 2 where d * 1 is the disk orthogonal to d 2 , d -2 , d x and
	d 1 ⊂ d * 1 ; d * 2 is the disk orthogonal to d 1 , d -1 , d x and d 2 ⊂ d * 2 (see Figure 2.4).
	Lemma 2.3.1. For any square-pyramidal disk arrangement, the mirror disks and the tan-
	gency disk are well-defined.
	Proof. We first prove the Lemma for the standard [D ⊠ ] -1 x (κ 1 ) which appears in the
	Figure 2.4.
	The orthogonality conditions of d * 1 imply that the boundary of d * 1 must be the
	circle with center (0, -1) For d * 2 , a disk orthogonal to d 1 2
	which gives that d * 2 is the half-space {x ≥ 0}.

•

Apollonian groups and packings 65

Corollary 3.7.2. The octahedral Apollonian group is isomorphic to

⟨C 123 , C 123 , C 123 , C 123 , C 123 , C 123 , C 123 , C 123 ⟩ < O ↑ 3,1 (Z[ √ 2])

where 3.7.

  Polytopal Descartes' Theorem, as can the generalizations of Soddy and Gosset,

	Guettler and Mallows, and Dias and Nakamura. Then, we shall use the Polytopal
	Descartes' Theorem to construct integral Apollonian packings based on the Platonic
	solids.

1

Chemistry Nobel Prize laureate in 1921 for discovering isotopes. the

  Let κ 1 , . . . , κ d+1 be the curvatures of d + 1 pairwise tangent d-balls of a polytopal d-ball packing B O d+1 . Then,

	4.1.2.
	By Prop. 3.6.1, both definitions are equivalent.
	Corollary 4.2.3. d+1
	i=1

  If there is an integral polytopal d-ball packing B T d+1 , B O d+1 or B C d+1 then the following Diophantine equations, respectively, have integer solutions.

	2.2, 4.2.3
	and 4.2.4 are integers, we obtain a geometric method to find solutions to three Dio-
	phantine equations.

Corollary 4.2.5. Let d ≥ 1.

  Such arrangement of three orthoplicial sphere packings {B O 4 , B ′ O 4 , B ′′ O 4 } will be called an orthoplicial trinity. Let {B O 4 , B ′ O 4 , B ′′ O 4 } be an orthoplicial trinity. Then, for any B ∈ {B O 4 ,B ′ O 4 ,B ′′ O 4 }, we have that B * = {B O 4 , B ′ O 4 , B ′′ O 4 } \ B. Proof. By Cor. 3.6.4, it is enough to prove the result in a particular case. Let us consider the standard orthoplicial sphere packing [B O 4 ] 1 2 given in Fig. 5.2. The dual of [B O 4 ] 1 2 can be split into two orthoplicial sphere packings, both obtained from [B O 4 ] 1 2

	Lemma 5.2.2.

  ⟨t 123 , t 123 , t 123 , t 123 , t 123 , t 123 , t 123 | t ijk := s ijk4 s ijk4 ⟩ In this case, the group isomorphism ϕ O : Γ O → A(B O

Theorem 5.3.1. Let

  Ω(B P ) be either a tetrahedral, an octahedral or a cubical Apollonian packing. There is an orthoplicial Apollonian packing Ω(B O 4 ) containing an Apollonian section arithmetically equivalent to Ω(B P ). Moreover, Ω(B P ) is integral if and only if Ω(B O 4 ) (Integrality condition for orthoplicial Apollonian packings). Let κ 1 , κ 2 , κ 3 and κ 4 be the curvatures of four pairwise tangent spheres of an orthoplicial sphere packing B O 4 . Then, Ω(B O 4 ) is integral if and only if κ 1 , κ 2 , κ 3 , κ 4 and T 3 (κ 1 , κ 2 , κ 3 , κ 4 ) are integers.Proof. Nakamura gave a proof in[START_REF] Nakamura | The local-global principle for integral bends in orthoplicial Apollonian sphere packings[END_REF] for all orthoplicial sphere packings Möbius equivalent to the standard [B O 4 ] 1 2 . Since the orthoplex is Möbius unique, Nakamura's lemma applies to every orthoplicial sphere packing. Proof of Theorem 5.3.1. Let B O 4 be the orthoplicial sphere packing obtained from [B O 4 ] 1 2

	is integral.
	For the integrality part, we need the integrality condition for orthoplicial Apol-
	lonian sphere packings.
	Lemma 5.3.1

  κ -3 are also the curvatures of six spheres of B ′ O 4 . Let κ 4 and κ -4 be the curvatures of the remaining spheres of B ′We also have that κ 1 , κ 2 , κ 3 , κ 4 are the curvatures of four pairwise tangent spheres of

			O 4 . By the antipodal relation, which
	holds for B ′ O 3 and B ′ O 4 , we have
	κ O 3 =	κ 1 + κ -1 2	= κ O 4	(5.6)
	B ′ O 4 . By combining (5.6) with Cor. 4.2.3 we obtain

Remerciements

Je remercie chaleureusement les (post)doctorant(

Chapter 3. Polytopal ball packings

We show in Fig. 3.10 the tetrahedral, cubical and dodecahedral Apollonian packings containing the sequence of perfect squares.

Integrality of the Platonic Apollonian packings

In the same paper where Soddy found the generalization of Descartes' Theorem in three dimensions [START_REF] Soddy | The Kiss Precise[END_REF], he noticed, by resolving the quadratic equation (4.25) appearing in the Descartes' Theorem for κ 4 , that if κ 1 , κ 2 , κ 3 are the curvatures of three disks in a Descartes configuration, such that

then all the curvatures in the Apollonian packing containing the Descartes configuration are integers. An Apollonian packing satisfying this property is said to be integral (Fig. 4.2). This simple but deep observation of Soddy opened several lines of research in number theory (see [START_REF] Fuchs | Counting problems in Apollonian packings[END_REF][START_REF] Fuchs | Local-global principles in circle packings[END_REF] for two excellent surveys). In this section, we apply the Polytopal Descartes' Theorem to obtain integrality conditions analogous to (4.33) for the Platonic Apollonian packings.

-3

Introduction

In [START_REF] Maehara | On soddy's hexlet and a linked 4-pair[END_REF], Maehara and Oshiro studied two necklace representations of the simplest link: the Hopf link (Fig. 6.1). Chen used the same two constructions to obtain graphs which can not be realized as the tangency graph of a sphere packing [START_REF] Chen | Apollonian ball packings and stacked polytopes[END_REF].

The first known necklace is the Soddy's Hexlet, a nice construction discovered by Soddy in the same paper where he introduced the integral Apollonian packings and the three-dimensional generalization of Descartes' theorem [START_REF] Soddy | The Kiss Precise[END_REF]. Soddy's Hexlet appears naturally as a subset of the classic Apollonian sphere packing [START_REF] Kontorovich | The local-global principle for integral soddy sphere packings[END_REF]. The second construction considered by Maehara and Oshiro is a standard orthoplicial sphere packing, and they used it to compute the ball number of the Hopf link, which is 8 [START_REF] Maehara | On configurations of solid balls in 3-space: Chromatic numbers and knotted cycles[END_REF]. As far as we are aware, the Hopf link is the only link whose ball number is known. We define the orthocubical shifts as the elements { µ x , µ y , µ z , µ -x , µ -y , µ -z } ⊂ SA(B ↑z O 4 ) corresponding to the cubical shifts after conjugation with ψ -1 C , where ψ C is the bijection mapping Σ C to Ω(B C 3 ) described in the proof of Prop. 5.

Chapter 6. Orthoplicial Apollonian packings and rational links

Conlusions: current and future work

unique, the Möbius spectra can be defined for any polyhedral graph. Möbius uniqueness implies that the Möbius spectra does not depend on the edge-scribed realization. It would be interesting to verify if the Möbius spectra is a complete invariant for edge-scribable Möbius unique polytopes and, in particular, for polyhedral graphs.

Question 2. Are there two combinatorially different edge-scribable and Möbius unique (d+ 1)-polytopes with the same Möbius spectra? In particular, are there two non-isomorphic polyhedral graphs with the same Möbius spectra?

Integral packings

Among the many corollaries of the Polytopal Descartes' theorem proved in Chapter 4, we obtained the integrality conditions needed to construct integral Apollonian packings based on the Platonic solids. Queries studied in the literature about the behavior of the integers (multiplicity, density, primes...) appearing in the integral tetrahedral Apollonian packings [START_REF] Fuchs | Counting problems in Apollonian packings[END_REF][START_REF] Fuchs | Local-global principles in circle packings[END_REF][START_REF] Bourgain | A proof of the positive density conjecture for integer Apollonian circle packings[END_REF][START_REF] Kontorovich | Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds[END_REF] , can be also considered for the Platonic Apollonian packings. For instance, by comparing the integrality conditions of tetrahedral (Eq. 4.33) and icosahedral Apollonian packings (Prop. 4.3.6), we find that every triple of integers which produces an integral tetrahedral Apollonian packing also produces a φ-integral icosahedral Apollonian packing (see Fig.

Diophantine equations

Besides the quadratic equations obtained by the Descartes' theorem and its generalizations, the curvatures of an integral Apollonian packing have been used to find integers solutions to other Diophantine equations [START_REF] Kocik | On a Diophantine equation that generates all integral Apollonian gaskets[END_REF][START_REF] Frisch | Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples[END_REF]. In Corollaries 4.2.5 and 6.4.2, we have presented two geometric methods based on integral polytopal dball packings to obtain solutions of several Diophantine equations. The first method follows from the Polytopal Descartes' theorem, and is obtained from the curvatures in the corresponding integral packing, as usual. The second method is slightly different, since the solutions do not come from the curvatures, but from the inversive coordinates of tangency points in a polytopal Apollonian packing. We believe that the parametrization given in Eq. (6.6) gives all the positive primitive solutions of the Diophantine equation in Cor. 6.4.2, which is equivalent to the following conjecture. Conjecture 4. The positive primitive integer solutions of x 4 + y 4 + z 4 = 2t 2 are in bijection with the set of first tangency points η p/q with p and q positive coprime integers.

Necklace representations and the ball number

Conlusions: current and future work

Seifert's algorithm [START_REF] Seifert | Über das geschlecht von knoten[END_REF] to orthocubical diagrams of rational links, we are able to construct polyhedral Seifert surfaces of links with a small number of planar triangles.

In Fig.

Software Polytopack

Besides the theoretical topics investigated in this thesis, we developed a software on Mathematica [57] that we called Polytopack [START_REF] Rasskin | Polytopack: a framework for polytopal sphere packings[END_REF] and used to plot all the 2D and 3D figures of this thesis. This helped us to better understand some objects, testing geometric properties on necklace representations, as well as the arithmetical behaviors of the curvatures of Apollonian packings. This software has been continuously improved through the last years, and it counts now with multiple options dealing Conlusions: current and future work

Appendices d

Name Notation Schläfli symbol Eigenvalues Multiplicities Midsphere ratio

Layer n Curvatures 1 1 κ P -φ 5 -φ -1 h 0 2 4 κ P -φ 5 -1 h 0 3 12 κ P -φ 4 + φ 2 h 0 4 24 κ P -φ 4 + φ h 0 5 12 κ P -φ 4 + 1 h 0 6 4 κ P -φ 4 + φ -1 h 0 7 24 κ P -φ 4 h 0 8 24 κ P -φ 3 + φ h 0 9 32 κ P -φ 3 + 1 h 0 10 24 κ P -φ 3 h 0 11 12 κ P -φ 2 + 1 h 0 12 24 κ P -φ 2 + φ -1 h 0 13 28 κ P -φ 2 h 0 15 24 κ P -φh 0 16 24 κ P -h 0 17 54 κ P 18 24 κ P + h 0 . . . . . . . . .