Arnaud Mortier

Finite-type 1-cocycles of knots given by Polyak-Viro formulas

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Finite-type 1-cocycles of knots given by Polyak-Viro formulas

Introduction

The study of the topology of the space of knots was initiated in 1990 by V.A.Vassiliev [START_REF] Victor | Cohomology of knot spaces[END_REF], who defined finite-type cohomology classes by applying ideas from the finite-dimensional affine theory of plane arrangements to the infinite-dimensional theory of long knots in 3-space. Here, of finite type means of finite complexity, in some sense, hence hopefully computable. Independently, outstanding general results on the topology of knot spaces have been obtained by A.Hatcher [START_REF] Hatcher | Spaces of Knots[END_REF], leading to the idea that higher dimensional invariants of knots -in particular, 1-cocycles -should capture information about the geometry of a knot (see [START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF]).

The zeroth level of Vassiliev's theory, known as finite-type knot invariants, has been extensively studied in the subsequent years [START_REF] Birman | Knot polynomials and Vassiliev's invariants[END_REF][START_REF] Kontsevich | Vassiliev's knot invariants[END_REF][START_REF] Goussarov | Finite-type invariants are presented by Gauss diagram formulas[END_REF][START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF]. However at the first level -that of 1-cocycles, only one example, in degree 3, has been proved to exist by D.Teiblum and V.Turchin, and then actually described by V.A.Vassiliev with a formula over Z 2 [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF][START_REF] Victor Turchin | Computation of the first nontrivial 1-cocycle in the space of long knots[END_REF]. Since then, no progress has been made, probably because of the technicity of Vassiliev's construction and the apparent difficulty of turning it into a systematic method: indeed, it involves singularity theory with differential geometric conditions. The theory seems to be stuck at this one question: is there a way to evaluate finitetype 1-cocycles combinatorially, i.e. without involving geometry?

Our main result is that the answer is yes. In [START_REF] Mortier | Polyak type equations for virtual arrow diagram invariants in the annulus[END_REF][START_REF] Mortier | Virtual knot theory on a group[END_REF], it was shown that Polyak-Viro's arrow diagram formulas [START_REF] Polyak | Gauss diagram formulas for Vassiliev invariants[END_REF] (the most compact combinatorial formulas to describe finite-type invariants) are the kernel of an explicit linear map with values in a space of degenerate arrow diagrams. In Section 1 we show that this map has a natural interpretation as a coboundary map in a simplicial model of the space of knots, adapted to the particular feature of Polyak-Viro's formulas that is counting subdiagrams. As a result, the 1-codimensional objects in this simplicial model are natural candidates to describe 1-cocycles in the space of knots.

Concretely, just like Polyak-Viro's invariants are defined via the finite set of crossings of a knot projection, the ways they are arranged (the Gauss diagram), and its power set (the subdiagrams, endowed with weights that are products of writhe numbers), we consider the finite set of germs of a loop in the space of knots endowed with a projection -that is, its local behaviour near each Reidemeister move, together with their subgerms, also weighted with the product of their writhe numbers. Cocycles defined via knot projections should vanish on the meridians of the 2codimensional strata defined by the higher order Reidemeister theorem [START_REF] Fiedler | A 1-parameter approach to links in a solid torus[END_REF][START_REF] Fiedler | Knot polynomials via one parameter knot theory[END_REF] -these strata also appear as a set of elementary moves in the study of surfaces embedded in 4-space, known as Roseman moves, in their "movie" version [START_REF] Carter | Surfaces in 4-space[END_REF][START_REF] Carter | Knotted surfaces and their diagrams[END_REF]. In Section 2, we explain how to write down the system of equations derived from those strata, and how to reduce it so as to make it computable.

In the last section, we compute that system of equations in degree 3 and study the properties of one of its solutions which we call α 1 3 . It is proved in that the evaluation of α 1 3 on the rotation of a long knot K around its axis is equal to -v 2 (K). This was conjectured to hold (up to sign) for the Teiblum-Turchin cocycle in [START_REF] Victor Turchin | Computation of the first nontrivial 1-cocycle in the space of long knots[END_REF]. Lastly, we prove that the reductions mod 2 of α 1 3 and the Teiblum-Turchin cocycle are equal by showing directly that α 1 3 mod 2 is of finite type. On the basis of these facts, we make the following conjecture.

Conjecture 1. Every 1-cocycle of knots defined by an arrow germ formula of degree n represents a finite-type cohomology class of degree no greater than n.

As an immediate consequence of this conjecture, one would obtain: Conjecture 2. The arrow germ formula α 1 3 is a realization of Teiblum-Turchin's cocycle over Z.

1 Knot invariants from a simplicial viewpoint

General principle

One of the most common ways for a knot theorist to define a knot invariant consists of two steps: first, start from a knot diagram and construct something from it, say an element of an abelian group A; then, prove that the result does not change when one performs Reidemeister moves. There is an obvious interpretation of this process in terms of cellular cohomology with coefficients in A. Indeed, one may think of the set of knot diagrams isotopic to a given one as a cell, bounded by (finitely many, two-sided) codimension 1 cells that are Reidemeister moves: associating something with every knot diagram amounts then to the choice of a 0-cochain, and by applying the Stokes formula one sees that this cochain defines a knot invariant if and only if it is a cocycle. Of course, this cellular complex is huge and there is no hope to compute its first coboundary map in general. However, assume that one has constructed a simple family of maps F from the set of knot diagrams to A, and wonders which linear combinations of these maps give knot invariants? The answer can be given by an adequate model for the coboundary map from the previous discussion. Namely:

1. Construct a set F 1 of objects that are likely to represent how the elements of F behave under Reidemeister moves.

2. For each element f 1 ∈ F 1 and each Reidemeister move D D ′ associate an element of A, denoted by f 1 , D D ′ .

3. Construct a linear map d : ZF → ZF 1 that satisfies the formula:

f (D ′ ) -f (D) = d(f ), D D ′ .
Here ZF is the Z-module freely generated by the elements of F . In other words, one has to build every ingredient in a Stokes formula, and prove that the formula is actually satisfied. There is a two-sided benefit from such a construction. First, the initial goal was achieved since computing the invariants coming from the family of maps F has been reduced to computing the kernel of an explicit matrix. Second, the elements of F 1 are now very good candidates to produce 1-cocycles of knots.

By 1-cocycle of knots, we mean a 1-cocycle in the space of all smooth embeddings S 1 ֒→ R 3 . The simplicial viewpoint can be extended one step further, thanks to the higher dimensional Reidemeister theorem that describes codimension 2 strata corresponding to the "simplest" degeneracies of Reidemeister moves (see [START_REF] Fiedler | A 1-parameter approach to links in a solid torus[END_REF][START_REF] Fiedler | Knot polynomials via one parameter knot theory[END_REF]). These strata will be described in Section 2.1. We now give our main example of a family of maps F that fits nicely into such a cohomological framework.

The example of arrow diagram invariants

In [START_REF] Mortier | Polyak type equations for virtual arrow diagram invariants in the annulus[END_REF] and [START_REF] Mortier | Virtual knot theory on a group[END_REF] it has been shown that Goussarov-Polyak-Viro's combinatorial formulas for Vassiliev invariants ( [START_REF] Goussarov | Finite-type invariants of classical and virtual knots[END_REF]) are the kernel of an explicit linear map. We show here that this map is a part of a 0-coboundary map in the spirit of the previous section. The framework is that of long knots and based Gauss diagrams, because they have easier combinatorics, and because it was the original settings for which V.A.Vassiliev defined his finite-type cohomology [START_REF] Victor | Cohomology of knot spaces[END_REF]. However, every statement can be adapted to other kinds of 1-dimensional knotted objects such as the virtual knots on a group introduced in [START_REF] Mortier | Virtual knot theory on a group[END_REF], or tangle diagrams in the disc -paying attention to symmetries whenever they may appear (see [START_REF] Östlund | A combinatorial approach to Vassiliev knot invariants[END_REF], Sections 2.2 and 2.4, and [START_REF] Mortier | Virtual knot theory on a group[END_REF], Section 4.1.2).

Notations and basic notions

Convention. When several incomplete diagrams are represented side by side in a picture, or in one and the same equation, it is to be understood that 1. Every unseen part -including missing decorations, such as local orientations -is the same for all diagrams.

2. The picture is valid no matter what are those unseen parts, unless otherwise specified in the caption. However, the visible parts cannot contain additional arrow ends or a missing point at infinity.

Notations for the 0-codimensional data

A (based) arrow diagram is an oriented line together with a finite number n of abstract oriented chords attached to it at 2n distinct points. We shall represent such a diagram as based on a circle with a distinguished point "at infinity". A (based) Gauss diagram is an arrow diagram in which every arrow has been decorated with a sign, + or -. Both types of diagrams are regarded up to positive homeomorphisms of the real line.

Fact. (see [START_REF] Louis | Virtual knot theory[END_REF]) Gauss diagrams are in 1-1 correspondence with (virtual) long knot diagrams in the plane R 2 up to diagram isotopy (and detour moves).

This means that a Gauss diagram represents exactly what we want to think of as a cell of maximal dimension. Consequently, we introduce the Q-space G freely generated by all Gauss diagrams and think of it as our space of 0-chains. It is graded by the number of arrows.

Gauss diagrams enjoy a set of R-moves, which are equivalent to the usual Reidemeister moves for knot diagrams. Their skeleton is depicted in Fig. 1; the signs and orientations have to follow some rules explained thereafter. An R 1 -move is the birth or death of an isolated arrow, as shown in Fig. 1 (top-left). There is no restriction on the direction or the sign of the arrow.

R 2 -moves

An R 2 -move is the birth or death of a pair of arrows with different signs, whose heads are consecutive as well as their tails (Fig. 1, top-right). Finally, define w(e) as the product of the signs of the two arrows at the boundary of e.

An R 3 -move is the simultaneous switch of the endpoints of three arrows as shown on Fig. 1 (bottom), with the following conditions:

1. The value of w(e)ε(e) should be the same for all three visible edges e. This ensures that the piece of diagram containing the three arrows can be represented in a knot-diagrammatic way without making use of virtual crossings.

2. The values of ↑(e) should be pairwise different. This ensures that one of the arcs in the knot diagram version actually "goes over" the others.

Remark 1.2. From our cellular viewpoint, the sign w(e)ε(e) gives a natural co-orientation of the 1-codimensional strata corresponding to R 3 moves. This "good" co-orientation is part of the reason why the 1-cocycle formulas we are going to find later are so simple.

Proof that our criterion for R 3 -moves is correct. Condition 2 is obviously necessary. Now assume that it is satisfied. It is enough to check only one case where Condition 1 is satisfied, and one case where it fails. Indeed, all possible cases are linked by an abstract connected graph (the cube of R 3 -moves, see Fig. 2), and the status of Condition 1 is preserved by walking along this graph. Namely, two adjacent triangles from Fig. 2 differ at the level of Gauss diagrams by switching two consecutive arrow ends, and changing the sign of the third arrow.

+ - + + - - Figure 2:
The cube of all eight possible local R-III situations and "edges" between them -the diagram lies on S 2 , and the "unbounded" region is the eighth triangle.

Any arrow diagram A defines a linear form on G by the Polyak-Viro formula [START_REF] Polyak | Gauss diagram formulas for Vassiliev invariants[END_REF]:

A, G = S(A), I(G) ,
where S(A) = sign(σ)A σ is the alternate sum of all (2 deg(A) ) completions of A into a Gauss diagram, I(G) is the formal sum of all (2 deg(G) ) subdiagrams of G, and , is the orthonormal scalar product with respect to the basis of Gauss diagrams. Roughly speaking, A, G counts the number of times the configuration A happens in G, with weights given by the product of the signs of the arrows involved.

The set of linear forms induced by arrow diagrams is our family of maps F in these settings (see Section 1.1). The Q-space freely generated by this set is denoted by A and regarded as a space of 0-cochains. A linear combination A ∈ A such that the map A, • is invariant under R-moves is called a (virtual) arrow diagram formula.

Notations for the 1-codimensional data

Definition 1.3 (germs, subgerms, partial germs). An i-germ, for i = 1, 2, 3, is an ordered couple (G 0 , G 1 ) of Gauss diagrams that differ by an R i -move. To prevent possible ambiguity, the edges involved in the R-moves are distinguished -in the pictures, we will represent this by little dots (that shall not be confused with the point at infinity, always flanked by an ∞ sign). An arrow adjacent to at least one distinguished edge is called distinguished as well.

Due to the nature of arrow diagram formulas (counting subdiagrams), we need to introduce the notions of subgerms and partial germs. Note that there is a natural correspondence between the arrows of the two diagrams in an i-germ (excluding those involved in the R-move in cases i = 1, 2).

A partial 3-germ (or simply partial germ) is the result of removing one distinguished arrow from a 3-germ. The only intact distinguished edge is still called distinguished (as well as the adjacent arrows).

A subgerm of a germ (G 0 , G 1 ) is the result of removing a (maybe empty) set of arrows from G 0 , together with their match in G 1 -in particular, no distinguished arrow can be removed in cases i = 1, 2. Also, in case i = 3, at most one distinguished arrow may be removed. Hence a subgerm is always either a germ itself, or a partial germ.

The notions of arrow i-germs, arrow subgerms and partial arrow germs are defined similarly, with no signs decorating the arrows.

The Q-spaces generated by i-germs (i = 1, 2, 3) and partial germs, modulo all the relations (x, y)+(y, x) = 0 in case i = 3, are respectively denoted by G I , G II , G ∆ and G Λ . The corresponding arrow diagram spaces are respectively denoted by A I , A II , A ∆ and A Λ .

The space A Λ is meant to be modded out by the triangle relations, that can be of two types. One of them is shown on Fig. 3, the other is obtained by reversing all the arrows in the picture. The quotient is denoted by A Λ / ∇ .

Finally, set

G 1 = G I ⊕ G II ⊕ G ∆ , and A 1 = A I ⊕ A II ⊕ A ∆ ⊕ A Λ / ∇ .
These are respectively our spaces of 1-chains and 1-cochains.

Remark 1.4. The triangle relations have appeared in several places in topology since M.Polyak found them at the beginning of his work on arrow diagrams enhanced by 3-vertices [START_REF] Polyak | On the algebra of arrow diagrams[END_REF] -see his review given in [START_REF] Polyak | Talk at Swiss Knots 2011 -"3 stories about[END_REF]. They also appear tautologically in V.A.Vassiliev's language in [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF]. Here, their meaning is the following: we have seen that in a cellular model, being a knot invariant should be equivalent to having zero coboundary, and obviously, zero and zero modulo something are not equivalent. However, partial arrow germs are not strictly a part of the cellular model that we discussed earlier: they are an artifact due to the particular settings of arrow diagram invariants that regard subdiagrams rather than simply diagrams. To recover an adapted cellular model, one should dilate the R 3 strata so that the three arrow-end switches happen separately. The oriented boundary of a dilated stratum is then exactly a triangle relator (this is explained in detail in [START_REF] Polyak | On the Casson knot invariant[END_REF]).

, , + ,

The boundary and coboundary maps

The boundary map ∂ is defined on the generators by

∂ : G 1 → G (G 0 , G 1 ) → G 1 -G 0 .
It obviously maps to zero the relators (x, y) -(y, x).

Definition 1.5. Let A be an arrow diagram. Consider the formal sum of all arrow germs and partial arrow germs (A 0 , A 1 ) such that A 1 (after forgetting that some edges are distinguished) is equal to A, and such that A 1 has no less arrows than A 0 . The class of this formal sum modulo the triangle relations is denoted by dA. This defines componentwise a linear map

d : A → A 1 , that splits into d I ⊕ d II ⊕ d ∆ ⊕ d Λ .

The pairing between G 1 and A 1

We mimic the definition of arrow diagrams as 0-cochains. The space G 1 ⊕ G Λ is endowed with the orthonormal scalar product with respect to the basis of germs, denoted by , . Again there is a map S defined componentwise by the formula ( * ∈ {I, II, ∆, Λ}):

S : A * → G * α → σ∈{±1} n sign(σ)α σ
, where α σ is the enhancement of α into a germ by sign decorations, and sign(σ) is the product of these signs. This map admits an adjoint for , , denoted by T , which forgets the signs of a germ and remembers only their product as a coefficient (see [START_REF] Mortier | Virtual knot theory on a group[END_REF], Section 4.1.2).

Finally, we introduce the linear map

I : G 1 → G 1 ⊕ G Λ
that sends a germ to the formal sum of its subgerms.

Definition 1.7. For α ∈ A I ⊕ A II ⊕ A ∆ ⊕ A Λ and γ ∈ G 1 , set α, γ = S(α), I(γ) = α, T •I(γ) .

Lemma 1.8. The value of the bracket α, γ only depends on the class of

α in A 1 .
This is a consequence of a deeper fact: let α = (A 0 , A 1 ) denote an arrow germ. Assigning sign decorations to the arrows as usual with a map σ : 1, . . . , deg(A 0 ) → {±1}, consistently for the two diagrams, one obtains a formal 3-germ α σ , which may or may not actually be a 3-germ (i.e. come from an R 3 move). It happens that the triangle relations and our pairing , detect that: Lemma 1.9. With the above notations, α σ is an actual 3-germ if and only if for every triangle relator ∇ (Fig. 3), ∇, α σ = 0.

Proof. Note that the definition of the pairing , obviously extends to formal 3-germs, so that the lemma makes sense:

∇, α σ = ∇, T •I(α σ ) .
We look only at partial 3-germs in the sum I(α σ ) since the other terms clearly do not take part in the computation. These can be grouped in triples, in which any two partial germs differ only by the arrow that they miss from an R 3 triangle (see an example on Fig. 4). Pick such a triple, say λ, and consider the sum ∇, T (λ) . It has 3 * 3 = 9 terms, and it is easy to see that either they are all zero, or two of them are non zero. In the latter case, we make the following observations.

+ + +

1. The two partial arrow germs non trivially involved appear in ∇ with opposite coefficientsindeed, the two at the bottom of Figure 3 cannot both appear in λ.

2. The product of the signs of the non-distinguished arrows is the same for all germs on Fig. 4.

3. For each partial germ (G 0 , G 1 ) from Fig. 3, one has ε(G 1 ) = +1. Thus ∇, T (λ) is zero if and only if computing ε times the product of signs of the visible arrows gives the same result for both relevant diagrams in λ, which is precisely a consequence of the fact that λ comes from an R 3 move (Section 1.2.1). This proves the "only if" part.

For the converse, pick a ∇ relation of degree two (i.e. without unseen arrows in Fig. 3). Then the only relevant terms in the sum I(α σ ) are those obtained by forgetting all arrows but two from the R 3 triple. There are only three such partial germs. The computation is the same as before, and the assumption that the bracket is zero allows one to conclude, using again the description of R 3 moves from Section 1.2.1.

The Stokes formula

Theorem 1.10 (Stokes Formula). For all A ∈ A and γ ∈ G 1 , dA, γ = A, ∂γ .

Remark 1.11. As an immediate corollary, we see that Ker d is exactly the set of arrow diagram formulasi.e. the elements of A that define knot invariants. A result of this kind was already proved by different means in [START_REF] Mortier | Polyak type equations for virtual arrow diagram invariants in the annulus[END_REF] and [START_REF] Mortier | Virtual knot theory on a group[END_REF]. It is easy to see that the map d constructed in these earlier works is isomorphic with our map d Λ . Also, let us mention that the important Lemma 3.2 from [START_REF] Mortier | Polyak type equations for virtual arrow diagram invariants in the annulus[END_REF] can be restated in the present language as

Ker d ∆ ⊂ Ker d Λ ∩ Ker d II .
Proof. By bilinearity it is enough to prove the formula for an arrow diagram A and a germ γ.

First assume that γ is a 3-germ. Then dA, γ = d ∆ A + d Λ A, γ . Choose a map σ that adds a sign to each arrow of A, and (consistently) to each arrow in each diagram in the sum dA -for this consistency to make sense, we use the particular representative of d Λ A that was constructed in Definition 1.5 before it was pushed modulo the triangle relations.

If γ = ( G0 , G1 ) is a partial subgerm of γ, then:

(dA) σ , γ = d Λ A σ , γ = A σ , G1 -G0 . (1)
Indeed, the first equality is obvious, and as for the second, 1. If A σ is different from both G0 and G1 , then both sides of the equality are zero.

2. If A σ ∈ G0 , G1 , then there is exactly one term in the sum d Λ A that coincides with either γ or -γ, and the sign is +1 if

A σ = G 1 , -1 otherwise.
Similarly, if γ = ( G0 , G1 ) is a subgerm of γ, then:

(dA) σ , γ = d ∆ A σ , γ = A σ , G1 -G0 . ( 2
)
Finally, notice that if G is a subdiagram of G 1 in which less than two arrows remain from the R 3 triple, then G is a subdiagram of G 0 as well and the corresponding contribution to A, ∂γ is 0. By definition of a subgerm, one never erases more than one arrow from the R 3 triple, so these diagrams do not contribute on the left-hand-side either.

It follows that the Stokes formula is the sum of the equations ( 1) and ( 2) (with both sides multiplied by sign(σ)) over all possible choices of σ and γ. Now assume that γ = (G 0 , G 1 ) is an i-germ with i = 1 or 2. We treat only the case i = 1, the other case is similar. Assume without loss of generality that G 1 has more arrows than G 0 . Then I(∂γ) is the formal sum of all subdiagrams of G 1 such that the distinguished arrow has not been removed. Indeed, all other subdiagrams of G 1 are in 1-1 correspondence with the subdiagrams of G 0 , and happen in I(∂γ) with opposite coefficients.

Again let σ be a particular completion of A into a Gauss diagram, and let γ = ( G0 , G1 ) be a subgerm of γ. We shall explain why

(dA) σ , γ = d I A σ , γ = A σ , G1 . ( 3
)
The first equality is obvious. As for the second, recall that d I A is the sum of all arrow 1-germs (A 0 , A 1 ) such that A 1 = A and A 1 has more arrows than A 0 . Hence

d I A σ , γ is equal to 1 if
A σ = G1 and to 0 otherwise. The second equality follows.

As before, the Stokes formula is the alternate sum of equalities (3) over all possible choices of σ and γ.

General method for producing 1-cocycle formulas

From the viewpoint of singularity theory, the Reidemeister moves are the 1-codimensional strata in a stratification of the space of knots induced by the choice of a projection R 3 → R 2 . Hence, for a linear combination of arrow germs (which is a 1-cochain defined as an intersection form with some of those strata) to define a 1-cocycle, it has to vanish on the meridians of the codimension 2 strata, i.e. the boundary of transversal 2-discs. The description of those strata, which are the "simplest" degeneracies of Reidemeister moves, is known as a higher order Reidemeister theorem [START_REF] Fiedler | A 1-parameter approach to links in a solid torus[END_REF][START_REF] Fiedler | Knot polynomials via one parameter knot theory[END_REF], but it also appears as a set of elementary moves in the study of surfaces embedded in 4-space, known as Roseman moves [START_REF] Carter | Surfaces in 4-space[END_REF].

Fortunately, we will see in Section 2.2 that most of these strata may be ignored for our purposes (in fact all but three of them), which gives the associated system of equations a reasonable size.

Codimension 2 strata and the associated equations

Let us recall the different types of strata, together with the corresponding meridians. In the description, we will say that two Reidemeister pictures are adjacent if they share exactly one common crossing. As usual, we use the notations of type "R-I" for Reidemeister moves of knot diagrams, and "R 1 " for R-moves of Gauss diagrams.

1.

Transverse intersection between two 1-codimensional strata. It is the situation where two Reidemeister moves are far from each other, so that one can choose in what order to perform them. These strata will be respectively denoted by , , 1. As we will see later (Proposition 2.8), R-I and R-II moves can be safely ignored in our theory. Hence, the heart of this equation is the fact of relating two Reidemeister III moves that share a common edge (as in the middle picture from Fig. 9). Note that Fig. 2 contains simultaneously all the possible local situations. Also, one can see on this picture the polyhedron which is dual to the cube that gave its name to the -equations [START_REF] Fiedler | A 1-parameter approach to links in a solid torus[END_REF][START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF].

R-I R-II R-III R-I R-II

R-III R-III R-II R-II

6.

Regular quadruple point -denoted by . It is the simultaneous degeneracy of two adjacent R-III moves. Note that instead of simultaneous degeneracies of Reidemeister pictures, one can think of , and as one branch sliding over (or under, or between) the branches of a Reidemeister move.

How to write down the equations associated with each stratum

We apply the same process that was used to show that the Polyak algebra is a complete set of equations for Goussarov-Polyak-Viro's invariants of virtual knots [START_REF] Goussarov | Finite-type invariants of classical and virtual knots[END_REF]. No distinction is made between a meridian and the corresponding formal sum of germs in G 1 .

I(m) = I(m; s)
where s runs over all subsets of the set of bystanders, and I(m; s) is the formal sum of subgerms extracted from m where exactly all bystanders that do not lie in s have been removed.

Lemma 2.3.

For any m and s as above, I(m; s) is the image under the map I of a linear combination of meridians.

Proof. One readily checks that

I(m; s) = s ′ ⊂s (-1) ♯s\s ′ I(m s ′ ),
where m s ′ is the meridian obtained from m by removing all bystanders that do not lie in s ′ .

The expressions I(m; s) are the analogue of the relations defining the Polyak algebra in [START_REF] Goussarov | Finite-type invariants of classical and virtual knots[END_REF]. Here it is possible to go a little further. Definition 2.4. We call a 1-cocycle equation any element of A 1 that is the image under T •I of a linear combination of meridians. By definition, an element α ∈ A 1 defines a 1-cocycle if and only if it satisfies (i.e. is orthogonal to) all 1-cocycle equations. By Lemma 2.3, the space of equations is exactly spanned by the elements of the form T (I(m; s)). Choose a meridian m of type or : the equations T (I(m; s)) have terms in only two different consecutive degrees, depending on whether or not an arrow from the R 3 -triple has been removed. Now let m ′ be the meridian obtained from m by changing the signs of all arrows involved the R 3 -move (and do so consistently in all germs of m); m ′ is indeed a meridian, by the criterion for R 3 -moves given in Section 1.2.1.

One sees that the homogeneous part of lower degree in T (I(m ′ ; s)) is equal to that of T (I(m; s)) (because those parts involve exactly two arrows from the R 3 triple), while their parts of higher degree are opposite. Hence

T (I(m; s)) = T (I(m; s)) + T (I(m ′ ; s)) 2 + T (I(m; s)) -T (I(m ′ ; s)) 2
is the decomposition of T (I(m; s)) in homogeneous parts, which proves the lemma for types and . The proof for the remaining types does not require more ideas.

Reduction of the set of equations

Before writing down the equations associated with all 2-strata, we are going to show that some equations, as well as some variables, may be ignored.

Keeping only one tetrahedron equation

In the same spirit as small "sets" of Reidemeister moves can generate all Reidemeister moves (as is shown in [START_REF] Polyak | Minimal generating sets of Reidemeister moves[END_REF]), one naturally expects some 2-meridians to express as linear combinations of others. Since the equations derived from are by far the most numerous and the most complicated, we try to get rid of them in priority. In the meridian from Fig. 10, one could change the orientation and the "level" (along the projection axis) of each strand. It makes a total of 48 combinatorially different meridians. Following [START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF], we call the one presented on Fig. 10 positive braid-like, because all visible crossings have a positive writhe and because the strand orientations do not forbid that this meridian happens for braids.

The following is proved in [START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF].

Lemma 2.7. The positive braid-like -meridian, together with the meridians of other types of 2-strata, generates all other -meridians.

While this lemma is inherent to the topology of the space of knots, all the simplifications made hereafter rely on the particular shape of the 1-cochains we are considering.

Getting rid of 1-germs and 2-germs

In the actual computation of the knot invariants given by arrow diagram formulas, it appears that R 1 -and R 2 -moves do not take part in the difficulty of the problem: indeed, the kernel of d I ⊕ d II is simply a subspace generated by some single arrow diagrams. Similarly, at the 1-codimensional level, all the variables related with R 1 -and R 2 -moves can be ignored.

Proposition 2.8. Any non trivial 1-cocycle presented by arrow germs admits a presentation using only arrow 3-germs and partial arrow 3-germs.

Proof. Let A be an arrow diagram. Denote by γ 1,2 (A) the set of all 1-and 2-germs involved in the sum dA. The collection of all such sets γ 1,2 (A) forms a partition of the set of all 1-and 2-germs. Now let α ∈ A 1 be an arrow germ formula, defining a non-trivial 1-cocycle. Claim: for any two arrow germs a, b lying in the same set γ 1,2 (A), one has α, a = α, b . Indeed, assume that -a and -b are far from each other, i.e. kill disjoint sets of arrows in A. Then the claim follows from α vanishing on the meridians of the strata , and . If, on the contrary, -a and -b both kill a common arrow in A, then it follows from α vanishing on the meridians of the strata and . As a consequence, there is a collection of rational numbers {α A }, almost all equal to zero, indexed by the set of arrow diagrams, such that α = α A dA + some arrow 3-germs and partial arrow 3-germs.

It follows now from the Stokes formula (Theorem 1.10), that the combination αα A dA still defines a 1-cocycle, which is cohomologous to that defined by α.

How to ignore some R 3 strata as well Proof. The claims are direct consequences of the fact that a 1-cocycle vanishes on the meridians of (point 1), (point 2), and (points 3 and 4). Indeed, once 1-germs, 2-germs and nonmonotonic germs have been removed, the homogeneous equations extracted from these meridians (Proposition 2.6) contain only one term left, and these one-term equations kill all diagrams of the indicated forms. The only non-trivial observation, for point 3, is that in a -meridian, the arrow born and killed by R 1 is always one of the two arrows that are distinguished in the monotonic partial subgerms of the R 3 -move.

Altogether, we have proved the following theorem.

Theorem 2.11. The linear system with ➺ as variables: arrow 3-germs and monotonic partial arrow 3-germs that do not satisfy any of the four conditions from Proposition 2.10, and ➺ as equations: all homogeneous equations extracted from the meridians of , and by Proposition 2.6, is a complete system for the 1-cohomology classes of knots given by arrow germ formulas.

Some remarks and questions

1-cocycles of virtual knots

Because it is based on a Gauss diagram construction, our method produces by nature 1-cocycles in the space of virtual knots -and hence, of course, of actual knots as well. Indeed, they are solutions to all Gauss diagrammatic equations associated with the meridians of 2-strata in the space of knots, regardless of whether the diagrams in those meridians actually correspond to knot projections.

In fact, arrow germ formulas for 1-cocycles of actual knots should only satisfy a subset of our equations, but it is very hard to determine that subset. An analog of this phenomenon can be observed for arrow diagram formulas: Polyak-Viro's formula for v 3 [START_REF] Polyak | Gauss diagram formulas for Vassiliev invariants[END_REF] is not an invariant of virtual knots, and as a result, the proof that it is actually an invariant for actual knots requires some special knowledge of arrow diagram identities -see [START_REF] Östlund | A combinatorial approach to Vassiliev knot invariants[END_REF].

Arrow germ presentations of the trivial 1-cocycle

By the Stokes formula (Theorem 1.10), all "derivatives" of linear combinations of arrow diagrams are trivial 1-cocycles. Question 2.12. Is the converse true? Namely, is it true that any arrow germ formula defining a trivial 1-cocycle is actually the derivative of a linear combination of arrow diagrams?

Although this would seem quite natural, it is not obvious -and in fact we have no clue whether it is true or not.

However, if it is not, then it would still be interesting to understand what are actually the "primitives" of those trivial cocycles. Can some combinations of them lead to new knot invariants?

Finite-type cohomology classes

Recall that the degree of an arrow germ (A 0 , A 1 ) is defined by max(deg(A 0 ), deg(A 1 )). We end this section with the most important question of this article.

Conjecture 2.13. Every 1-cocycle presented by an arrow germ formula of degree n is of finite-type, of degree at most n.

Section 3 will provide a number of clues in favor of this conjecture. In general, here is already one reason to believe that arrow germs and Vassiliev's theory should be related: although they are most of the time thought of via planar projections, one should not forget that local writhes are actually well-defined signs associated with the two desingularizations of a double point, for singular knots in any oriented 3-manifold (see [START_REF] Fiedler | A small state sum for knots[END_REF], Lemma 1). In other words, (products of) writhe signs are natural co-orientations of Vassiliev's singular strata: it makes it possible to understand what is the oriented boundary of an arrow germ formula in the first stage of Vassiliev's filtration (see Section 3.3).

Teiblum-Turchin's cocycle

The first -and, so far, the only -example of a finite-type integral 1-cocycle of knots is the socalled Teiblum-Turchin cocycle v 1 3 . No formula is known to actually evaluate it, except mod 2 [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF][START_REF] Victor Turchin | Computation of the first nontrivial 1-cocycle in the space of long knots[END_REF]. In this section, we compute completely the linear system described in Theorem 2.11 in homogeneous degree three. Then we study one of the solutions, called α 1 3 , which we conjecture is equal to the Teiblum-Turchin cocycle. Two properties are proved to support this conjecture: first, the evaluation of α 1 3 on the rotation of a long knot K around its axis gives the Casson invariant v 2 (K), which is conjectured to hold for v 1 3 in [START_REF] Victor Turchin | Computation of the first nontrivial 1-cocycle in the space of long knots[END_REF]; second, we show that α 1 3 mod 2 is of finite type, which implies the equality

α 1 3 = v 1 3 over Z 2 .

Arrow germ formulas of degree three

Note that the lowest degree of a germ involved in the meridian of a -stratum is 4, hence we are left with the study of the two strata and i.e. the cube and tetrahedron equations.

The cube equations

There are 6 × 2 3 meridians of type from the viewpoint of Gauss diagrams. The factor 6 corresponds to the six diagrams on Fig. 11 (the lower row of the picture shows as an example the meridian associated with the first diagram), and each factor 2 is a binary choice, that is actually not essential: For each picture, there are exactly two choices of signs that make the R 2 -and R 3 -moves possible, and this choice has no influence in the resulting equation.

2. For each picture, say , one can switch the positions of the two arrow ends that are alone -in the example, . Again the associated equations are identical.

3. Reversing all the arrows in the pictures yields the same equations, with all arrows reversed. In particular, note that the set of variables that we have not banned from the system (Theorem 2.11) is stable under the arrow-reversing operation.

The equations of degree 3 derived from the six meridians on Fig. 11 are respectively given in Fig. 12 -all of them are homogeneous parts of degree 3 of T (I(m; s)) with s = ∅ (Definition 2.2). Note that substitutions can be made so that equations d) and e) contain both only (four) partial germs. Also, note that c) contains three different equations, indexed by the choice of a point at infinity. Remark 3.1. Our notational convention for partial arrow germs (Fig. 12) is not anecdotic. It corresponds to the co-orientation of R 3 -strata given by the sign w(e)ε(e) (see Remark 1.2). Arrow germ formulas of degree n simply count the intersection index (with R 3 strata endowed with this coorientation), weighted by the sign of n -2 arrows.

The tetrahedron equations

A Gauss diagrammatic description of all meridians coming from the positive braid-like quadruple point has been written in [START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF] (pp. 35-46). One readily sees that all degree 3 equations coming from those meridians are simple linear combinations of the cube equations (Fig. 12), except for the one corresponding to global type II, with point at infinity 2, and global type IV, with point at infinity 3. ( [START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF], pp. 37-38 and 41-42 ), which are obtained from each other by reversing the orientation of all the arrows. They are shown in Fig. 13, using the same notational convention as in Fig. 12. It is proved in [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF] that the group of finite-type integral 1-cocycles of degree 3 is isomorphic to Z. Hence, if Conjecture 2.13 is true, one cannot expect more than essentially one non-trivial solution to the system of equations on Figs.12 and 13. Therefore, we shall only extract and study one such solution.

Theorem 3.2. The arrow germ formula α 1 3 on Fig. 14 defines a 1-cocycle of knots.

Proof. One readily checks that it satisfies all equations from Figs.12 and 13 as well as the ones obtained from them by reversing the orientations of the arrows. Since the arrow germs involved in α 1 3 do not take part in any other meridian than those of types and , the theorem follows. For any long knot K, there is a loop rot(K) canonically defined by rotating K positively around its axis. We repeat in Fig. 15 the diagrammatic description of rot(K) given in [START_REF] Fiedler | Quantum one-cocycles for knots (v2)[END_REF]. To convince oneself that this loop amounts to one rotation around the axis, it can be helpful to think of the two isotopic bands (or framed long knots) at the top-left of Fig. 15. On the other hand, v 2 (K) denotes the Casson invariant of K, which is the only non-trivial Vassiliev invariant of degree 2; it is given by the following arrow diagram formula ( [START_REF] Polyak | Gauss diagram formulas for Vassiliev invariants[END_REF], Theorem 1 and remark on p.451 ):

K K K K K = no Reidemeister move
v 2 (K) = , K .
Theorem 3.3. For all knots K, α 1 3 (rot(K)) = -v 2 (K). In [START_REF] Victor Turchin | Computation of the first nontrivial 1-cocycle in the space of long knots[END_REF], V.Turchin conjectures that the Teiblum-Turchin cocycle v 1 3 satisfies the above equation (with the opposite sign). His conjecture would follow from our Conjecture 2.13 and this theorem.

Proof. In all this proof the dot on each diagram stands for the point at infinity. R 3 -moves only happen in the two horizontal arrows on Fig. 15. For the top arrow, it is easy to see that all of them have two arrows from the R 3 -triple in position , hence they cannot be seen by α 1 3 . In the bottom horizontal arrow on Fig. 15, there is exactly one R 3 -move for each crossing in the initial diagram of K. Moreover, the R 3 -triple is in position 1.

p for each arrow p in position p in the initial diagram, and 2. p for each arrow p in position p .

The formula α 1 3 is oblivious of R 3 -moves in situation 2. As for situation 1, the last three germs of Fig. 14 do not participate, and the first germ counts the number of arrows q in position q p with respect to p and the point at infinity. Obviously arrows q in such a position can only come from the initial diagram (all other arrows, that are due to the arc slide over K, have their head "immediately after" the point at infinity). Hence the theorem is proved modulo 2. We have to understand the signs.

For each couple p, q as above, corresponding to the situation s r q p , α 1 3 counts the sign

w(p) × w(q) × w(r) × ε,
where w(•) is the sign of an arrow, and ε is equal to +1 if the arrows p and r cross each other before the move, -1 otherwise (by the convention on Fig. 12). Since v 2 (K) counts w(p) × w(q) for all such couples, the claim is that one always has w(r) × ε = -1. By a diagram isotopy, one can assume that from the beginning, every crossing in the diagram of K has its two branches oriented "to the right" as in Fig. 16 (this only serves to reduce the number of situations to consider). Then there are only two possible situations, depending on w(p); they are depicted on Fig. 16, which shows the diagrams right before the R 3 -move is performed. In each of these situations, one has w(r) = +1 and ε = -1, hence w(r) × ε = -1 as announced. Proof of the corollary (the notations are introduced thereafter, after [START_REF] Victor | Cohomology of knot spaces[END_REF]). It appears in the proof of Theorem 3.4 that the reductions mod 2 of α 1 3 and v 1 3 have the same principal part in σ 3 \ σ 2 (Fig. 19; see [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF], Stabilization formula over Z 2 and Proposition 6 ). Since the reduced homology group Hω-3 (σ 2 ; Z 2 ) is trivial [START_REF] Victor | Cohomology of knot spaces[END_REF], it follows that α 1 3 and v 1 3 are cohomologous over Z 2 .

Remark 3.6. This gives an answer to a question that was asked by T.Fiedler ten years ago [5]: yes, there is a description of v 1 3 mod 2 that does not involve differential geometric aspects. However, it will appear in the proof of the theorem that the homological puzzle inside Vassiliev's resolved discriminant still contains some differential geometric pieces.

The rest of this section is devoted to the proof of Theorem 3.4. Let us recall how Vassiliev's finite-type cohomology classes are defined -after [START_REF] Victor | Cohomology of knot spaces[END_REF] and [START_REF] Victor | Complements of discriminants of smooth maps: topology and applications[END_REF]. The space of all smooth immersions R → R 3 is denoted by K. To think of it as a manifold, Vassiliev constructs approximations of this space by finite-dimensional affine spaces of polynomial immersions, and shows that part of his construction stabilizes as the dimension tends to infinity. We will denote here this finite dimension of K by the letter ω. The space K has a natural stratification induced by the degree of non genericity of the immersions. Non generic immersions form the discriminant Σ of K.

The crucial observation is made that using Alexander duality, an n-cocycle in the space of knots -that is, in K \ Σ -is the linking form with some (ωn -1)-cycle in the one-point compactification of Σ, relative to the compactification point. From then on, the idea of Vassiliev is to measure the complexity of those cycles:

1. Each elementary part of Σ is artificially dilated, i.e. replaced with a cell of a higher dimension, which depends essentially on its complexity in the stratification.

2. This new space made of dilated cells -called a resolution of the discriminant, is endowed with a filtration, also related with the initial complexity of each cell in the stratification.

3. An n-cocycle in the space of knots is said to be of finite type no greater than k if the associated (ωn -1)-cycle in Σ has a lift in the k-th stage of the filtration of the resolution.

How to relate this with the formula defining α 1 3 ? By nature, arrow germ formulas are intersection forms with certain relative 1-codimensional chains in the space (K, Σ). The boundary of these chains in Σ, of course, does not have itself a boundary in Σ. But it does have a non trivial boundary in the resolution of Σ. The question is, can this "artificial" boundary be pushed deeper into the filtration so as to vanish (see Fig. Note that [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF] adopts the opposite approach: D.Teiblum and V.Turchin had found a candidate for the "principal part" of a cycle, i.e. a relative homology class in σ 3 \ σ 2 , and V.A.Vassiliev computed its successive boundaries in the associated spectral sequence until he found an actual formula for the 1-codimensional chain on the left of Fig. 17.

Vassiliev calculus

In practice, each cell (J-block in [START_REF] Victor | Complements of discriminants of smooth maps: topology and applications[END_REF]) in the resolution of Σ is graphically represented by a based circle marked with finitely many distinguished points; a point can be distinguished in two ways that are not mutually exclusive: being marked with a star, or being linked by a chord to one or more other distinguished points. Roughly speaking, the chords and stars indicate where the singular knots that respect the diagram should have double points and vanishing derivative.

Two essential quantities are associated with such a cell: its complexity which is its degree in the filtration, and its dimension. Denote by |A| the number of points in the circle that bound at least one chord; by ♯A, the number of connected components of the abstract graph of chords; by b, the number of stars.

➺ The complexity i of a cell is given by the formula:

i = |A| -♯A + b.
➺ Its dimension is given by: dim = (ω-3i)+(number of distinguished points)+(number of chords + number of stars -1).

The first summand (ω -3i) is the dimension of the set of knots in K that respect the chord diagram. It explains the choice of i as a complexity. The first two summands together, (ω -3i) + (number of distinguished points), give the dimension of the set of knots in K that respect the chord diagram up to equivalence, which is defined as usual by positive homeomorphisms of the based circle. Finally, the last summand corresponds to an artificial simplex whose vertices are the chords and the stars of the diagram. Roughly speaking, it is this simplex that enables high dimensional chains to "live" in the highly degenerate parts of Σ. Remark 3.7. Despite the presence of ω -. . . in the dimension formula, one must not forget that those cells are not embedded in K, hence they do not have a codimension there. With one exception though: the cells of complexity 1 are naturally homeomorphic with actual parts of Σ (of codimension 1 in K), since the artificial simplex has dimension 0 in this case. The crucial point to make actual computations is that the boundary of such a cell is not in general a combination of other similar cells. Rather, it is a combination of subvarieties of cells, sometimes defined by conditions of (differential) geometric nature. This phenomenon requires additional notations on cell diagrams. In general, to compute the dimension of a cell diagram enhanced with geometric conditions, one first computes the dimension of the ambient cell, and then the codimension induced by the additional conditions. There are two basic additional decorations on a cell diagram: oriented chords, and dashed chords. They indicate that, inside the ambient cell, we only consider the knots with a classical crossing -respectively, double point -as indicated by the chord. If the endpoints of this chord do not coincide with any other particular point of the cell diagram, then this additional condition is of codimension 0 -respectively, 1.

First let us recall the non-geometric types of degeneracies.

1. A chord turns into a dashed chord. It corresponds to the loss of one dimension in the artificial simplex -the knots, however, still have the corresponding double point. Note that this may or may not involve a -1 jump in the filtration degree.

2. An oriented chord turns into a dashed chord. It means that the branches of the corresponding crossing tend to coincide and form a double point.

3. An edge in the chord diagram is shrunk to a pointi.e. two distinguished points tend to coincide. The special rules that accompany this degeneracy is described in [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF] (pp. 44-45 ).

We now give in Fig. 18 a list of examples of exceptional situations that involve differential geometry. It may not be exhaustive in general, but it covers every typical case in our computations related to α 1 3 . The superscripts over the chord diagrams in Fig. 18 respect the notations from [START_REF] Victor | Combinatorial formulas for cohomology of spaces of knots[END_REF] and should be read respectively as follows (f denotes in each case a parametrization of the knot).

1. f ′ (a) and f ′ (b) have colinear projections to R 2 with non-negative coefficients and, moreover, the partial derivative at b with respect to the third coordinate is greater than that at a. With those indications, one can now readily check that the following pieces (Figs.19 to 23) fit well into a cycle in the resolved discriminant, and that the part in C ω-2 (σ 1 ) (Fig. 23) is indeed the boundary of the chain presentation of α 1 3 (Fig. 14). The only non-trivial point that remains to be made clear is why the last diagram on Fig. 21 , this degeneracy would involve too large a jump in dimension: the cusp at the star point should be tangent to the other branch that passes through this point, a condition of codimension 1 in a cell of dimension ω -3. ). The superscripts here mean that we consider only singular knots with this prescribed behaviour on each side of the stratum

+

Figure 1 :

 1 Figure 1: R-moves for Gauss diagrams (see below the rules for the decorations)

R 3 -moves Definition 1 . 1 (

 11 The sign ε and the co-orientation of R 3 moves). In a classical Gauss diagram of degree n, the complementary of the arrows is made of 2n oriented components. These are called the edges of the diagram. In a diagram with no arrow, we still call the whole circle an edge.Let e be an edge in a Gauss diagram, between two consecutive arrow ends that do not belong to the same arrow. Put η(e) = +1 if the arrows that bound e cross each other -1 otherwise , and let ↑(e) be the number of arrowheads at the boundary of e. Then define ε(e) = η(e) • (-1) ↑(e) .

Figure 3 : 1 . 2 . 2

 3122 Figure 3: One triangle relation for partial arrow germs -the other is obtained by reversing the orientation of every arrow

Remark 1 . 6 .

 16 The reason why d I and d II do not count the germs (A 0 , A 1 ) with deg(A 1 ) < deg(A 0 ) is the following. By essence, when an arrow diagram looks at a Gauss diagram, what it sees is all of the subdiagrams of G. Now in the boundary of an i-germ (i = 1, 2), the Gauss diagram with less arrows happens twice as a subdiagram, with opposite coefficients, whence arrow diagrams do not see it at all.

Figure 4 :

 4 Figure 4: One possible triple in the sum I(α σ )

  , the meridian of a stratum of type looks like the following:

Figure 5 : 2 .Figure 6 : 3 .

 5263 Figure 5: Meridian of a transverse intersection between 1-codimensional strata

Figure 7 : 4 .Figure 8 : 5 .

 7485 Figure 7: Meridian of a cusp with a transverse branch

Figure 9 :

 9 Figure 9: Meridian of a regular tangency with a transverse branch -the cube equation

Figure 10 :

 10 Figure 10: The meridian of a (positive, braid-like) quadruple point -the tetrahedron equation. Each step is made of two R-III moves.

Definition 2 . 2 .

 22 Let m ∈ G 1 be a meridian. The arrows corresponding to unseen crossings in Figs.5 to 10 are called bystanders. Roughly speaking, they are arrows which never become distinguished. Fix a meridian m ∈ G 1 . The sum I(m) naturally splits as

Definition 2 . 5 .

 25 The degree of an arrow germ α = (A 0 , A 1 ) is defined by deg(α) = max(deg(A 0 ), deg(A 1 )).

Proposition 2 . 6 .

 26 The subspace of 1-cocycle equations admits a family of homogeneous generators (in the sense of the degree defined above). More precisely, any homogeneous part of an equation T (I(m; s)) is itself a 1-cocycle equation.Proof. For meridians of type , , , and , the equations T (I(m; s)) are themselves already homogeneous.

4 .(

 4 Specific to 3-germs) -a distinguished arrow is isolated on one side of the germ and, if one removes it, the two remaining distinguished arrows are in a position to be killed by R 2 .

Figure 11 :

 11 Figure 11: Six diagrams representing the essentially different cube meridians, with one example

  equations a) and b) again.

Figure 12 :Figure 13 :Figure 14 : 3 3.2 α 1 3 (

 12131433 Figure 12: Half of the degree 3 cube equations -for the other half, reverse all the arrows

Figure 15 :

 15 Figure 15: The loop rot(K)

Figure 16 : 3 . 3 The reduction of α 1 3 mod 2 3 over the integers. Theorem 3 . 4 .Corollary 3 . 5 . α 1 3 and v 1 3

 16332334353 Figure 16: Two possible situations if the branches are directed to the right near every crossing

17

 17

Figure 17 :

 17 Figure 17: Finite-type 1-cocycles of degree 3

Figure 18 :

 18 Figure 18: Exceptional degeneracies accompanied with differential geometric conditions

2 .

 2 f ′ (a) and f ′ (b) have colinear projections to R 2 with non-negative coefficients.

3 .

 3 f ′ (a) is directed "up" (in the direction of projection, towards the eye).

4 . 5 .

 45 f ′ (a) and f ′ (b) have opposite directions. Diagram on the left: the projection to R 2 of f ′ (c) is a linear combination of the projections of f ′ (a) and f ′ (b), with non-negative coefficients. Diagrams on the right: see point 2.

  does not have a piece of boundary in the cell . The reason is that because of the superscript c b,a a,b

Figure 19 :Figure 20 :Figure 21 :Figure 22 :Figure 23 :

 1920212223 Figure 19: The Alexander dual cycle to α 1 3principal part in C ω-2 (σ 3 \ σ 2 )

  Thanks to Proposition 2.8, we may forget about the R 1 -and R 2 -moves involved in all meridians. It makes some equations particularly simple, and enables us to kill yet another family of germs. From now on, we will always consider arrow germ formulas that are only made of 3-germs and partial germs, called arrow 3-germ formulas. Also, to remove the indeterminacy due to the triangle relations, we consider only those partial arrow germs that switch an arrowhead and an arrowtail, called monotonic partial (arrow) germs. It is easy to see that they form a basis of A Λ / ∇ .

	Convention-Definition 2.9.

Proposition 2.10. A germ cannot participate in an arrow 3-germ formula if it satisfies any of the following conditions.

1. A non-distinguished arrow is isolated (i.e. in a position to be killed by R 1 ).

2. Two non-distinguished arrows are in a position to be killed by R 2 .

3. (Specific to partial germs) -a distinguished arrow is isolated on one side of the germ.

Acknowledgements

I wish to express my full gratitude to Seiichi Kamada for his invitation to the Osaka City University Advanced Mathematical Insitute where this work was done, and his help through the Japanese administration. I am also grateful to Victoria Lebed for her inestimable support.

I thank Thomas Fiedler, Michael Polyak and Victor Turchin for fruitful discussions.


Floriane Anstett-Collin

Thierry Mara

Michel Basset

Application of global sensitivity analysis to a tire model with correlated inputs

Keywords: Global sensitivity analysis, correlated parameters, polynomial

When a vehicle equipped with tire is manoeuvred on the ground, the tires are submitted to a number of forces -longitudinal force when driving or braking torque is applied to the wheel and/or lateral force when the wheel is steered to turn at a corner. Pacejka model describes these forces that represent the reaction of the road onto the tire. This nonlinear model depends on correlated parameters such as the friction coefficient, the vertical load, the cornering stiffness, . . . which have to be identified from some measurements.

The sensitivity of Pacejka model to these correlated parameters are studied using an approach based on polynomial chaos. It consists in decorrelating the parameters using the Nataf transformation and then, in expanding the model output onto polynomial chaos. The sensitivity indices are then obtained straightforwardly from the algebraic expression of the coefficients of the polynomial expansion.

chaos expansion, tire model

Introduction

In the automotive and aeronautical fields, modelling the tire/road interface is fundamental. Indeed, the tire model is one of the major elements to integrate into a ground vehicle or aircraft model, as the tires are the only contact surface with the road. When a vehicle equipped with tires is manoeuvred on the ground, they are subject to a number of forces -longitudinal force when driving or braking torque is applied to the wheel and lateral force when the wheel is steered to turn at a corner. The tire models available in the literature describe the efforts and moments corresponding to the road reaction of the tire (see for instance [START_REF] Pacejka | Tyres and vehicle dynamics[END_REF][START_REF] Rajamani | Vehicle dynamics and control[END_REF][START_REF] Gipser | Dynamical tyre force response to road unevennesses[END_REF][START_REF] Blundell | Multibody systems approach to vehicle dynamics[END_REF]).

One of the most famous model was proposed by Pacejka ( [START_REF] Pacejka | Tyres and vehicle dynamics[END_REF]) and is often used ( [START_REF] Bakker | Tyre modelling for use in vehicle dynamics studies[END_REF]) nowadays by industrials (Michelin, SAE, Adams tire software, ...). This model is nonlinear, complex and depends on a certain number of parameters (friction coefficient, vertical load, side slip angle, ...) that can be obtained from experimentation. Unfortunately, experimental data are often sparse or incomplete, especially in the aircraft domain and their measurements are very expensive. On the one hand, some parameters only have a negligible influence on the model response and therefore, do not need to be determined accurately. On the other hand, some others are relevant for the model response and thus influence its uncertainty significantly. These parameters may require additional measurement effort in order to be estimated with relatively high accuracy. In order to prepare and plan future experiments, it is necessary to perform a sensitivity analysis of the Pacejka model.

Several studies have focused on the global sensitivity analysis of nonlinear models with independent inputs (for instance, [START_REF] Cukier | Nonlinear sensitivity analysis of multiparameter model systems[END_REF][START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF][START_REF] Saltelli | A quantitative model independent method for global sensitivity analysis of model output[END_REF][START_REF] Frey | Identification and review of sensitivity analysis methods[END_REF][START_REF] Saltelli | Sensitivity analysis for importance assessment[END_REF][START_REF] Saltelli | Sensitivity Analysis in Practice. A Guide to Assessing Scientific Models[END_REF][START_REF] Blatman | Sparse polynomial chaos expansion and adaptive stochastic finite element using a regression approach[END_REF]). They rely on variance-based sensitivity indices also known as Sobol' indices ( [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF]).

They measure the contribution of input parameters to the model response variance. Such a contribution can be due to a parameter alone or to a group of parameters. Sobol' indices are unique and easily interpretable when the parameters are independent. But, it is more challenging when they are dependent. Indeed, if a function (i.e. model response) structurally depends on parameter p 1 and not on p 2 , and if p 1 and p 2 are correlated, classical Sobol' indices will lead to the conclusion that both are relevant inputs. Even though actually p 2 is influential only because of its correlation with p 1 .

It may be of high interest to distinguish whether an input is relevant regarding its correlations with the other ones [START_REF] Jacques | Sensitivity analysis in presence of model uncertainty and correlated inputs[END_REF]. This information may be of great help for the experimenter in order to guide future experiments.

Mara and Tarantola ( [START_REF] Mara | Variance-based sensitivity indices for models with dependent inputs[END_REF]) have derived a set of variance-based sensitivity indices to cope with parameters dependency. They also proposed a computational method for their estimation. The proposed approach relies on the use of polynomial chaos expansion ( [START_REF] Wiener | The homogeneous chaos[END_REF]) in conjunction with a Gram-Schmidt based decorrelation procedure. They also indicate that Nataf ([16]) or Rosenblatt ([17]) transformations can be used.

This study is part of a french national project involving several industrial partners (Airbus, Messier-Dowty, Turbomeca, etc) and research centers, in particular the Modélisation Intelligence Processus Systèmes (MIPS) laboratory. The aim of the project is to take advantage of performances of the modern simulation tools by providing new models and developing tools which allow the simulation of systems and physical phenomenon in the aeronautical fields. More precisely, the laboratory works to develop models for simulating the tyre-road interaction characteristics with respect to the aircraft run types. In this framework, the aim of the paper is to analyze the well-known Pacejka tire model, which is a basic function of the Magic Formula ( [START_REF] Pacejka | Tyres and vehicle dynamics[END_REF]).

Pacejka model is widely used in the automotive and aeronautical fields. The model depends on correlated parameters (friction coefficient, cornering stiffness, vertical load, ...) which must be identified from measurement data.

The most relevant parameters on the lateral force are highlighted, using the approach proposed in [START_REF] Mara | Variance-based sensitivity indices for models with dependent inputs[END_REF].

The paper is organised as follows. Section 2 presents the Pacejka tire model. Section 3 recalls the expression of the Sobol' sensitivity indices for model with independent parameters and their estimation based on polynomial chaos. Section 4 presents the approach used to study the sensitivity for models with correlated parameters. Then, the sensitivity analysis of the Pacejka tire model is performed.

Pacejka tire model

Tires in motion on the ground are subject to a number of forces. For example, a longitudinal force is developed when driving or braking torque is applied to the wheel. A lateral force appears when the wheel is at an angle or when it is steered to turn at a corner. The model considered in this study accounts for the longitudinal force (driving or braking force) and the lateral force. In the present study, we will exclusively focus on the model expression for the lateral force developed in the case of a cornering manoeuvre in steadystate condition, as represented in figure 1. In pure cornering condition, an interpolation function, called Magic Formula ([1]), is proposed for the lateral force. The Pacejka model presented here is a basic function of the Magic Formula. In this case, the lateral force is given by:

F y = D sin[C arctan(B(α + S h ) -E(B(α + S h ) -arctan(B(α + S h ))))] + S V (1)
The model output of interest F y is the lateral force, road reaction in the lateral direction. The parameter α is the side slip angle, that is the angle between the wheel plane and the wheel direction of motion (see figure 1).

The parameter D represents the maximum value that F y can reach and C, E, S h and S v are empirical fitting parameters. Figure 2 depicts the relationship between the lateral force F y and the side slip angle α. In figure 2, the product B × C × D, corresponding to the cornering stiffness, is the slope at the origin.

B = K DC , D = -3.4 × 10 -7 F 2 z + 0.74F z . ( 2
)
where K is the cornering stiffness and F z the vertical load. The ratio D/F z represents the lateral friction coefficient µ that characterizes the road state.

The uncertain parameters F z , α, C and K are independent. The ultimate aim of this study is to perform the sensitivity analysis of F y to the parameter set (B, C, D, α) because the model response directly depends on these four macro-parameters. But, let first analyze the contribution of the independent parameters (α, F z , K, C) on F y .

Data collected from the aeronautical field experimentation during the MIPS project framework and specifications from manufacturer leads to the distributions of the parameters C, α, K, F Z , given in Table 1. All parameters follow a normal distribution with mean and standard deviation given in Table 1. Table 1: Distribution of the independent parameters

Parameters

The next section recalls the expression of the Sobol' indices for models with independent parameters.

Sensitivity analysis for models with independent inputs

Sobol' indices

Let y = f (p) be a square integrable function of a set of n independent random inputs p = (p 1 , . . . , p n ) ∈ R n . Sobol' proved that f can be expanded as a sum of orthogonal functions of increasing dimensionality:

f (p) = f 0 + n i=1 f i (p i ) + n j>i f ij (p i , p j ) + . . . + f 1•••n (p 1 , • • • , p n ) (3)
and that this decomposition is unique ( [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF]).

Thanks to the orthogonality of the functions in equation ( 3), it is straightforward to decompose the variance of y, denoted V y :

V y = n i=1 V i + n j>i V ij + . . . + V 1•••n (4)
with:

V i 1 ...is = V ar(f i 1 ...is ) (5)
where V ar is the variance operator.

As explained in [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF], the Sobol' sensitivity indices are obtained by renormalising (5) with the total variance V y :

S i 1 ...is = V i 1 ...is V y , 1 ≤ i 1 ≤ . . . ≤ i s ≤ n. (6)
S i 1 ...is represents the amount of V y due to the interaction between (p i 1 , . . . , p is ).

In particular, the first-order sensitivity index, denoted S i , that represents the individual contribution of the parameter p i to the variance of y, writes:

S i = V i V y (7)
Finally, we define the total sensitivity index of parameter p i , denoted ST i , that includes its individual as well as its collective contributions to V y as follows: 3) onto orthogonal polynomials known as polynomial chaos (PC). Indeed, it was shown in [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansion[END_REF][START_REF] Rahman | Global sensitivity analysis by polynomial dimensional decomposition[END_REF] that the sensitivity indices could be evaluated as analytical expressions of the PC coefficients. Hence, the cost of computing the sensitivity indices is the one of evaluating the PC coefficients and requires only N model evaluations.

ST i = i∈u i ⊆{1,...,n} S u i (8) As a result, S i ≤ ST i ∈ [0, 1]. A parameter is irrelevant if ST i = 0

Polynomial chaos expansion (PCE)

In [START_REF] Wiener | The homogeneous chaos[END_REF], it was shown that the homogeneous chaos expansion could be used to approximate any function in the Hilbert space of square-integrable functions. Therefore, the model output y can be decomposed as follows:

y = +∞ j=0 α j Φ j (p) (9)
where the multivariate polynomial Φ j of degree j is given by the tensor product of the corresponding one-dimensional Hermite polynomials φ a j k :

Φ j (ξ) = n k=1 φ a j k (p k ) (10)
with a j k ∈ N the degree of φ a j k , such that |a j | = n k=1 a j k . The theorem of Cameron and Martin ([20]), ensures that an expansion as (9) converges in the L 2 -sense. The convergence rate is optimal when the parameters are normally distributed. For other parameters distribution, the authors in [START_REF] Xiu | The Wiener-Askey polynomials chaos for stochastic differential equations[END_REF] introduced the generalized polynomial chaos to ensure an optimal convergence rate (Legendre polynomials for uniform distribution, Jacobi polynomials for Beta distribution, . . . ).

In practice, the PC expansion ( 9) is truncated up to a finite degree d and the number of coefficients in the expansion equals:

M + 1 = (n + d)! n!d! (11)
The optimal degree d could be selected by incrementing its value until a target accuracy, for instance the determination coefficient R 2 , is reached.

However, this can lead to a large number N of model evaluations to compute the PC coefficients when increasing d because N has to be greater than M .

To overcome this problem, an adaptive scheme has been proposed in [START_REF] Blatman | Efficient computation of global sensitivity indices using sparse polynomial chaos expansions[END_REF] to build up a sparse PC expansion.

The deterministic PC coefficients (i.e. the α j 's) are the unknowns and several non-intrusive approaches are proposed in the literature to compute them.

They can be classified as regression techniques (e.g. [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansion[END_REF][START_REF] Berveiller | Stochastic finite element: a non intrusive approach by regression[END_REF][START_REF] Sudret | Application of a stochastic finite element procedure to reliability analysis[END_REF]) or projection techniques (for instance [START_REF] Rahman | Global sensitivity analysis by polynomial dimensional decomposition[END_REF][START_REF] Crestaux | Polynomial chaos expansion for sensitivity analysis[END_REF]). In the case of the regression method, it is generally advisable to use an over-sampling to determine the sample size N , resulting in a least squares solution for the over-determined system. It is proposed to use N = 2(M +1) in [START_REF] Eldred | Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design[END_REF] and N = (n-1)(M +1), in [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansion[END_REF], stating that taking more points does not improve the accuracy of the results. In light of the dependence of M on the PC order and the number n of parameters, the PC representation will be computationally efficient when small values of N and d are sufficient for an accurate representation of the random variables.

The computation of Sobol' indices is straightforward once computed the PC coefficients. Indeed, let define I k 1 ,...,ks the set of multi-indices j such that:

I k 1 ,...,ks = {0 ≤ a j k ≤ d, a j k = 0 ∀k ∈ {1, . . . , n}\{k 1 , . . . , k s }} (12)
The first-order sensitivity index S i of parameter p i is estimated as follows:

Ŝi = j∈I i α 2 j E(Φ 2 j (p i )) M j=1 α 2 j E(Φ 2 j (p)) (13)
where E is the expectation operator and the set I i corresponds to the polynomials depending on p i only, with the exception of all the others. Higher order sensitivity indices can be obtained in the same manner:

Ŝi 1 ,...,is = j∈I i 1 ,...,is α 2 j E(Φ 2 j (p i 1 , . . . , p is )) M j=1 α 2 j E(Φ 2 j (p)) (14)
The total sensitivity index is given by:

ŜT i = j∈I i + α 2 j E(Φ 2 j (p i )) M j=1 α 2 j E(Φ 2 j (p)) (15)
with I i + the set of multi-indices j defined by:

I i + = {0 ≤ a j ≤ p, a j k = 0, ∀k = i} (16)
The set I i + corresponds to polynomials Φ i depending on parameter p i and possibly on the other parameters.

It can be worth noting that sensitivity indices can be computed from the PC coefficients with almost no additional cost. Indeed, only elementary mathematical operations are needed to compute the indices from the PC coefficients.

Sensitivity of Pacejka model to

(α, F z , K, C)

Numerical details

Consider the set of independent parameters p = (α, F z , K, C). Samples of parameters are generated using the well-known Latin Hypercube Sampling (LHS) [START_REF] Mckay | Evaluating prediction uncertainty in simulation models[END_REF][START_REF] Constantine | Sparse pseudo spectral approximation method[END_REF]. Alternative sampling methods exist in the literature (maximin LHS, low discrepancy sequences [START_REF] Niederreiter | Random number generation and quasi-Monte-Carlo methods[END_REF][START_REF] Helton | Survey of samplingbased methods for uncertainty and sensitivity analysis[END_REF], ...) but LHS is particularly well adapted to create correlated samples as shown in [START_REF] Iman | A distribution-free approach to inducing rank correlation among input variables[END_REF]. Besides, it is easy to implement.

As these parameters are normally distributed, Hermite polynomials are employed. The number of parameters is n = 4 and a third degree PC expansion is investigated (d = 3). Consequently, the number of terms in the decomposition (number of unkowns) is M + 1 = 35, according to Eq.( 11). The 35 coefficients α j are determined by least-squares regression after evaluating the model with a LHS of size N = 4096. With a third degree PC, the determination coefficient is of 0.9994. Moreover, the relative error between the model response computed using Pacejka model directly vs using PC decomposition does not exceed 4%, showing good accuracy of the results. To avoid overfitting, the variances of the Pacejka model response and of the PCE are compared (the ratio is of 0.9786). This shows that a third degree PC is sufficient here.

The sensitivity indices are computed using equations ( 13)- [START_REF] Wiener | The homogeneous chaos[END_REF]. The results are summed up in Table 2. The side slip angle is the most influential parameter on the lateral force of Pacejka model, followed by F z and K, while C is not relevant. Consequently, the fitting parameter C can be set at a nominal value of its interval of variation, with no consequence on the variance of F y . As a result, the parameter F z determines the maximum value that the lateral force F y can reach for a given side slip angle α while K determines the lateral deflection of the tyre. This can explain the relative importance of these parameters.

Results and discussion

These indices are consistent with the results expected in the tire domain.

Note that, the sum of the first-order indices is about 0.95 which means that the contribution due to the interactions is low or put in other words, the model is almost additive.

However, the lateral force depends directly on the macro-parameters (B, C, D).

Hence, the analysis of the contribution of the dependent parameter set (B, C, D, α)

is of high interest for the designer. The next section is focused on sensitivity analysis of models with correlated parameters.

Sensitivity analysis for models with correlated parameters

Let us consider now a model f with non normally distributed and correlated parameters. We assume that the parameters are defined by a Gaussian copula characterized by a correlation matrix ρ assumed known. We further assume that the parameters margins h i (margin of p i ) and the associated cumulative density functions (CDF) H i are also given.

In [START_REF] Mara | Variance-based sensitivity indices for models with dependent inputs[END_REF], it is shown that several PC expansions (and consequently ANOVA decompositions) of the model response can be performed by firstly decorrelated the inputs with the Nataf transformation. These PCEs lead to the definition of variance-based sensitivity indices for models with correlated inputs.

Given that Nataf transformation yields several independent parameters sets ( [START_REF] Lebrun | Do Rosenblatt and Nataf isoprobabilistic transformations really differ?[END_REF]), several PCEs can be derived and several sensitivity indices be defined. This is explained in the next subsections.

Nataf transformation

The Nataf transformation ( [START_REF] Liu | A multivariate distribution models with prescribed marginals and covariances[END_REF][START_REF] Li | Nataf transformation based point estimate method[END_REF][START_REF] Noh | Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula[END_REF]) allows the transformation from original space to mutually independent standard normal one, under Gaussian copula assumption for the joint distribution of the random input vector [START_REF] Lebrun | An innovating analysis of the nataf transformation from the copula viewpoint[END_REF].

It does not require the joint probability density function (PDF) of the parameters. However, the marginal PDF of each parameter and the correlation matrix must be known, which is often the case in engineering.

The Nataf transformation is divided into two steps. The first one consists in renormalizing each non normal correlated variable p i into a normal correlated variable z i . This renormalization is a CDF matching condition:

z i = Ψ -1 (H i (p i )), i = 1, . . . , n (17)
where Ψ -1 is the inverse CDF of a standard normal variable. Note that z i contains the same information than p i .

The correlation matrix ρ ′ of z is symmetric definite positive and can be decomposed as:

ρ ′ = RR T ( 18
)
where R is the lower triangular matrix obtained from Cholesky decomposition of ρ ′ .

The second step of the Nataf transformation consists in transforming the correlated normal variables z into uncorrelated standard normal variables u:

u = R -1 z. (19)
It is worthwhile to note that R -1 is also lower triangular. As a consequence, u 1 is proportional to z 1 and so equivalent to p 1 . The new variable u 2 is a linear combination of z 1 and z 2 but is not correlated to u 1 (i.e. p 1 ), so u 2 represents p 2 without its correlation with p 1 . In the same manner, u 3 can be interpreted as the residual part of p 3 not due to (p 1 , p 2 ) and so on... Finally, u n is the proper residual part of p n uncorrelated with the other parameters.

It can be noted that the computational cost of the Nataf transformation is related to the dimension of the correlation matrix and is

1 3 n 3
, where n is the number of parameters.

Variance-based sensitivity indices

Now that we have obtained a set of independent random variables u = {u 1 . . . , u n }, their variance-based sensitivity indices can be computed via PCE, as explained in section 3.2. According to the previous discussion, we can infer that:

• (S 1 , ST 1 ) the correlation ratio and total effect of u 1 are those of p 1 .

These indices are also called full marginal and total contribution of p 1 to the variance of y because they include both its correlated and uncorrelated contributions. Indeed, p 1 is correlated to all the other parameters {p 2 , . . . , p n }.

• (S 2 , ST 2 ) the correlation ratio and total effect of u 2 are those of p 2 without its correlative contribution with p 1 . In the sequel, we will use the following notations (S 2-1 , ST 2-1 ).

• ...

• (S n , ST n ) the correlation ratio and total effect of u n are those of p n without its correlative contribution with the other parameters. They represent p n 's proper contributions and are denoted (S u n , ST u n ).

The sensitivity indices so defined are so interpreted because of their order in the set. Indeed, the above discussion is true for the canonical order, that is, p = {p 1 , p 2 , . . . , p n }. For a different ordering, the new sensitivity indices are interpreted in the same way. For instance, with the set {p 2 , . . . , p n , p 1 }, (S 2 , ST 2 ) and (S u 1 , ST u 1 ), among others, can be computed. The complete set (S i , ST i ) and (S u i , ST u i ) can be computed by circularly permuting the parameters order prior to the Nataf transformation.

The sensitivity indices so defined rank within [0, 1]. If ST u i = 0 but ST i is high, then it means that p i is only relevant because of its correlations with the other variables. In a sense, this means that the uncertainty of p i can be ignored because its impact on the model response is already embedded in the other inputs uncertainties. On the opposite, if ST i = 0 but ST u i is high, which occurs in presence of negative correlations, p i is a parameter to account for in the model.

ρ = α B C D         1
0.00 0.00 0.00 0.00 1 -0.39 -0.35 0.00 -0.39 1 0.00 0.00 -0.35 0.00 1

        α B C D (20)
We first start with the parameters set ordering (α, D, B, C). Using the approach described in the previous section, the variance-based sensitivity indices have been computed. The results are summed up in Tables 3 and4

Results and discussion

As previously, the results show that α has the highest sensitivity index and therefore is the most relevant parameter. As compared to the previous analysis, its sensitivity indices remain unchanged since it is not correlated to the other parameters. The following most relevant parameter is D. Since its contribution is mainly due to its uncorrelated part ( Ŝu D = 0.44 and Ŝc D = -0.14) and D = µF z , D has almost the same influence as the independent parameter F z on F y . As explained previously, D, and so F z , determines the maximum value of the force F y , for a given side slip angle α. This explains the relative high contribution of these parameters to the lateral force F y . However, there is also a contribution of its correlative part with B, which cannot appear in the first study.

The contribution of the uncorrelated part of B is Ŝu B = 0.18 and the one of its correlated part is Ŝc B = -0.15. It can be worth noting that the correlations being negatives, the sensitivity indices due to the correlation are also negatives. These indices are almost equal, but opposite in sign, leading to a small full marginal index ( ŜB = Ŝu B + Ŝc B = 0.03). This analysis shows the influence of the correlative part of the stiffness factor B to D and consequently to the vertical load F z . This cannot be highlighted from the previous analysis of the set of independent parameters (α, F z , K, C).

It is the same case for the parameter C ( Ŝu C = 0.07, Ŝc C = -0.06 and ŜC = 0.01). Its influence is low but not negligible, contrary to the previous analysis of independent parameters. This is due to the fact that it is correlated to B that contributes significantly to the output variance.

As previously, the total indices ŜT i are close to the marginal indices Ŝi , showing that there are few interactions between the parameters and their influence is mainly due to their individual contribution.

It has been shown that α is the most influential parameter. Lateral force is, in general, a nonlinear function of the slip angle α (figure 2). The relationship between lateral force and slip angle is initially linear with a constant slope determined by the cornering stiffness. In this region, the tread band is supposed to be in complete adhesion with the road surface. As slip angle grows, eventually the force starts to saturate due to the limited friction on the road, entering the nonlinear region and a sliding zone appears. The limit of handling is defined by the maximum available lateral force µF z . Here, this occurs for α ≈ 15 • .The sliding zone increases with the slip angle up to full sliding (see figure 3).

In the following, sensitivity analysis has been performed for the adhesion zone defined by α ∈ [0; 15 • ] and for the sliding zone defined by α ∈ [15

• ; 20 • ].
The other parameters are kept in their initial variation interval. This leads to the sensitivity indices Ŝα = 0.5871 in the adhesion zone and Ŝα ≈ 0 in the sliding zone. It shows that α is more influential in the adhesion zone.

In the sliding zone, F y is almost constant with respect to α, explaining the insignificant influence of α.

In order to plan the experiments for cornering manoeuvre, this study has shown that it is important to sufficiently excite the side slip angle and the vertical load which influences the stiffness factor, especially in the adhesion For the adhesion zone, the 95% confidence interval of the mean lateral force

Fy is computed and is of 43.5%. Simulations have been made with hypothetic reductions of the variation interval of α. The results are presented in Table 5.

Interval reduction of α 95% confidence interval of F y 5% 32.98% 10% 31.88% 15% 31.5% It reveals that a reduction of 15% of the variation interval of α implies a reduction of 12% of the confidence interval of Fy .

On the other hand, the previous study has shown that the second most influent parameter is D, followed by B. These parameters are dependent on F z and K. Hypothetic reductions of the variation interval of F z have been made combined with the 15% reduction of α. Then, reductions of the interval of K have been made combined with a 15% reduction of F z . The results are shown in Table 6.

Finally, reducing the variation interval of C does not lead to a significant reduction of F y , since C is very few influent on F y . In conclusion, acting on the most influent parameters allow to reduce the confidence interval of the mean lateral force of almost the half of its initial value.

The next step will be to extend the sensitivity analysis of the

Figure 1 :

 1 Figure 1: Pure cornering -General view of the tire deformation in the contact patch

αFyFigure 2 :

 2 Figure 2: Lateral force vs. side slip angle

  Normal distributions: N (mean; standard deviation) 7208 × 10 5 ; 9.7010 × 10 4 ) F z N (9.0034 × 10 4 ; 5.5803 × 10 3 )

4. 3 .

 3 Sensitivity analysis of Pacejka model to (α, B, C, D) 4.3.1. Numerical details Let us consider now the set of parameters (α, B, C, D). It is worth noting that the analysis that follows is performed with the same sample generated in section 3.3. Analysis of the parameters sample provides the following correlation matrix:

Figure 3 :

 3 Figure 3: Lateral force vs. side slip angle

  and does not interact with the other parameters if ST i = S i . The previous sensitiv-

ity indices can be calculated using Monte Carlo simulations (

[START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF]

) or Fourier analysis (

[START_REF] Saltelli | A quantitative model independent method for global sensitivity analysis of model output[END_REF]

). Using Monte Carlo simulations with a sample size of N for a model with n parameters leads to N ×2 n evaluations of the model function to compute all the indices. This can be very costly to manage. An alternative consists in casting the orthogonal functions in expansion (

Table 2 :

 2 First-order and total sensitivity indices

	α	F z	K	C
	Ŝα	ŜFz	ŜK	ŜC
	0.45 0.31 0.17 0.02
	ŜT α	ŜT Fz	ŜT K	ŜT C
	0.49 0.33 0.21 0.02

  .

	α	D	B	C
	Ŝα	ŜD-α	ŜB-αD	Ŝu C
	0.45	0.30	0.11	0.07
	Ŝu α	ŜD	ŜB-D ŜC-DB
	0.45	0.30	0.11	0.07
	Ŝα-BC	Ŝu D	ŜB	ŜC-B
	0.46	0.44	0.03	0.03
	Ŝα-C	ŜD-Cα	Ŝu B	ŜC
	0.45	0.30	0.18	0.01

Table 3 :

 3 First-order sensitivity indices

	α		D	B	C
	ŜT α	ŜT D-α	ŜT B-αD	ŜT	u C
	0.50	0.32	0.15	0.09
	ŜT	u α	ŜT D	ŜT B-D	ŜT C-DB
	0.50	0.32	0.15	0.09
	ŜT α-BC	ŜT	u D	ŜT B	ŜT C-B
	0.50	0.44	0.07	0.04
	ŜT α-C	ŜT D-Cα	ŜT	u B	ŜT C
	0.50	0.32	0.23	0.02

Table 4 :

 4 Total sensitivity indices

Table 5 :

 5 Hypothetical interval reductions of α

  Pacejka model, taking into account the dynamic characteristic of the tire, during transient-Interval reduction of F z 95% confidence interval of F y

	5%	29.39%
	10%	28.5%
	15%	28%
	Interval reduction of K 95% confidence interval of F y
	5%	24.4%
	10%	23.6%
	15%	22.8%

Table 6 :

 6 Hypothetical interval reductions of F z and K state, when driving cases as taxiing are considered.