Seiichi Kamada
email: skamada@sci.osaka-cu.ac.jp

Ocami

Kokoro Tanaka
email: kotanaka@u-gakugei.ac.jp

The shadow nature of positive and twisted quandle cocycle invariants of knots

Keywords: quandles, multi-term distributive (co)homology, quandle cocycle invariants, positive quandle cocycle invariants, twisted quandle cocycle invariants, shadow colorings

Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations -namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.

Introduction

A quandle is a set Q endowed with binary operations1 ⊳ and ⊳ satisfying

(a ⊳ b) ⊳ c = (a ⊳ c) ⊳ (b ⊳ c), (1) (a ⊳ b) ⊳ b = (a ⊳ b) ⊳ b = a,
(2) a ⊳ a = a.

(3)

A group with the conjugation operation a ⊳ b = b -1 ab is an important example. Quandles were introduced in [START_REF] Joyce | A classifying invariant of knots, the knot quandle[END_REF][START_REF] Matveev | Distributive groupoids in knot theory[END_REF] as an efficient tool for constructing knot

over all crossings of D, where ε( ) is the sign of (cf. Figure 2). Axioms (4)-( 5) were chosen so that the the multiset of ω-weights { W ω (D, C) | C ∈ C Q (D) } yields a knot invariant, called a quandle cocycle invariant. This construction admits several variations. Instead of quandle 2-cocycles, they use some other cohomological data, which we now recall. Together with a quandle Q, consider a Q-module4 -that is, a set M endowed with two maps ⊳, ⊳ : M ×Q → M5 satisfying Axioms (1)-(2) for all a ∈ M, b, c ∈ Q. The simplest examples are Q itself, and a one-element set (called a trivial Q-module), with evident module operations. Take also an abelian group A. Denote by C k (M, Q, A) the abelian group of maps from M × Q ×k to A, and put6

(d k l φ)(m, a 1 , . . . , a k+1 ) = k+1 i=1 (-1) i-1 φ(m ⊳ a i , . . . , a i-1 ⊳ a i , a i+1 , . . . , a k+1 ), ( 7
) (d k r φ)(m, a 1 , . . . , a k+1 ) = k+1 i=1 (-1) i-1 φ(m, a 1 , . . . , a i-1 , a i+1 , . . . , a k+1 ). ( 8
)
Maps d k l and d k r turn out to be anti-commuting differentials. Hence for all α l , α r lying in Z (or, more generally, in a commutative ring

R if A is an R-module), (C k (M, Q, A), α l d k l -α r d k r ) is a cochain complex. Denote by C k Q (M, Q, A
) the subgroup of maps vanishing on all tuples (m, a 1 , . . . , a k ) with a i = a i+1 for some i. Both d k l and d k r restrict to C k Q (M, Q, A), and thus so do their linear combinations. The cocycles / coboundaries / cohomology groups of ( ]). Note that a trivial M can be safely excluded from consideration (and from our notations), since M × Q ×k ∼ = Q ×k ; we will often talk about (M-)shadow cocycles / coboundaries / cohomology groups if M is non-trivial.

C k Q (M, Q, A), α l d k l -α r d k r ) receive the adjective • quandle if α l = α r = 1 ([FRS95, CJK + 03]); • positive quandle if α l = -α r = 1 ([CG14]); • (t-)twisted quandle if α l = 1, α r = t, and A is a Z[t ±1 ]-module ([CES02]); • two-term distributive in the general case ([PS14, Prz11
Positive ( [START_REF] Cheng | Positive quandle homology and its applications in knot theory[END_REF]) and twisted ([CES02]) variations of quandle cocycle invariants using positive and, respectively, twisted quandle 2-cocycles with trivial M are recalled in Section 2. Shadow invariants based on shadow quandle 2-cocycles are also reviewed; together with arc colorings, these use region colorings by elements of M, respecting the rule on Figure 1 B . In literature, mostly the case M = Q was considered formally and thoroughly ([RS00, CKS01, Kam02]); the general case seems to be folklore. In this paper, carefully chosen Q-modules allow us to present twisted (Section 3) and positive (Section 5) quandle cocycle invariants as shadow invariants. This instantly gives a proof of their invariance, avoiding technical verifications. Moreover, all results established for shadow invariants can now be applied in the positive and twisted cases. Such results, discussed in Section 4, include the equality of invariants obtained from cohomologous cocycles; a certain symmetry of the invariants; some restrictions on the values of weights one can encounter; higherdimensional generalizations; and shadow7 versions. In Section 6, the shadow ideas lead to a sharpening of twisted quandle cocycle invariants for multi-component links.

Shadow, positive, and twisted quandle cocycle invariants

In this section we recall three variations of the quandle cocycle invariant construction. From now on, fix a quandle (Q, ⊳) and an abelian group A.

-cocycle of Q -that is, a map ω : M × Q × Q → A satisfying ω(m, a, b) + ω(m ⊳ b, a ⊳ b, c) + ω(m, b, c) = ω(m ⊳ c, a ⊳ c, b ⊳ c) + ω(m, a, c) + ω(m ⊳ a, b, c), (9)
ω(m, a, a) = 0. (10)

Definition 2.1. The (shadow) ω-weight of a (Q, M)-colored knot diagram (D, C, C sh ) is the sum

W sh ω (D, C, C sh ) = a b m ω(m, a, b) - a b m ω(m, a, b) (11)
taken over all crossings of D.

Theorem 1. For any Q-module M, m ∈ M, and M-shadow quandle 2-cocycle ω of Q, the multiset

{ W sh ω (D, C, C sh ) | (C, C sh ) ∈ C sh Q,M (D), C sh (∆ ex (D)) = m }
depends only on the underlying knot, and not on the choice of its diagram D.

The (extremely rich) knot invariant thus obtained are called shadow quandle cocycle invariants, or simply shadow invariants.

Another variation uses a positive quandle

2-cocycle of Q -that is, a map ω : Q× Q → A satisfying (5) and ω(a, c) + ω(a ⊳ b, c) = ω(a ⊳ c, b ⊳ c) + ω(a, b) + 2ω(b, c). ( 12
)
Definition 2.2. The (positive) ω-weight of a Q-colored knot diagram (D, C) is the sum

W pos ω (D, C) = a b ε pos ( )ω(a, b), (13)
where the sign ε pos ( ) is defined on Figure 3 (here the checkerboard coloring of the diagram's regions is used, with the exterior region declared white).

ε pos ( ) + - Figure 3: Positive quandle cocycle invariants Theorem 2 ([CG14]). For any positive quandle 2-cocycle ω of Q, the multiset { W pos ω (D, C) | C ∈ C Q (D) } yields a knot invariant.
The knot invariant obtained is referred to as positive quandle cocycle invariant.

In order to present the third generalization of quandle cocycle invariants we are interested in, some technical but intuitive definitions are necessary.

Definition 2.3.

• The index9 i(∆) of a region ∆ in a knot diagram D is the algebraic intersection number for D and a path γ connecting ∆ to the exterior region ∆ ex (D). Sign conventions are indicated in Figure 4 A ; γ and D are supposed to have a finite number of simple transverse intersections.

• The source region ∆( ) of a crossing in D is defined in Figure 4 B 1011 .

• The index of a crossing is set to be i(

) = i(∆( )).
The index of a region/crossing does not depend on the choice of path γ satisfying the description above, and is thus well defined.

∆ ∆ ex (D) γ D + D + D - i(∆) = +1+1-1 = 1 A ∆( ) B a m m + 1 C Figure 4

: Regions and indices

Until the end of Section 4, suppose A to be a module over a commutative ring R, and fix an α ∈ R * . Take an α-twisted quandle 2-cocycle of

Q -that is, a map ω : Q × Q → A satisfying (5) and ω(a ⊳ c, b ⊳ c) -αω(a, b) -ω(a ⊳ b, c) + αω(a, c) + (1 -α)ω(b, c) = 0. ( 14
) Definition 2.4. The (twisted, or α-twisted) ω-weight of a Q-colored knot diagram (D, C) is the sum W tw ω (D, C, α) = a b ε( )α -i( ) ω(a, b) (15)
taken over all crossings of D. (Recall that ε( ) is the usual sign of , cf. To prove Theorems 2 and 3, one could check directly that Reidemeister moves and induced coloring changes preserve the positive/twisted ω-weight, as it was done in [START_REF] Cheng | Positive quandle homology and its applications in knot theory[END_REF][START_REF] Carter | Twisted quandle homology theory and cocycle knot invariants[END_REF]. Here we prefer a more conceptual solution: in the following sections, the multisets from these theorems are presented as shadow invariants.

Twisted quandle cocycle invariants via shadows

The first ingredient we need is the Q-module Z with m ⊳ a = m + 1. For this Q-module, the region coloring rule from Figure 1 B specializes to the one from Figure 4 C . Comparing it with the definition of the index of a region, one proves Lemma 3.1. Consider a (Q, Z)-coloring of a knot diagram D such that the exterior region receives color 0. Then the color of any region ∆ coincides with i(∆).

The region coloring from the lemma will be denoted by C ind : Regions(D) → Z. Remark that it is compatible with any arc coloring.

Next, we show how to transform a twisted quandle 2-cocycle into a (non-twisted) quandle 2-cocycle, at the cost of introducing our non-trivial Q-module Z.

Lemma 3.2. A map ω : Q × Q → A is an α-twisted quandle 2-cocycle if and only if the map ω α : Z × Q × Q -→ A, (m, a, b) -→ α -m ω(a, b) (16)
is a shadow quandle 2-cocycle.

Proof. Conditions (5) for ω and (10) for ω α clearly coincide. Further, condition (9) for ω α reads

α -m ω(a, b) + α -(m+1) ω(a ⊳ b, c) + α -m ω(b, c) = α -(m+1) ω(a ⊳ c, b ⊳ c) + α -m ω(a, c) + α -(m+1) ω(b, c),
which, due to the invertibility of α, is the same as (14).

We further compare α-twisted ω-weights with shadow ω α -weights:

Lemma 3.3. For any Q-colored diagram (D, C) and any α-twisted quandle 2cocycle ω, one has

W tw ω (D, C, α) = W sh ωα (D, C, C ind ).
Proof. It follows from the definitions and lemmas above.

Putting everything together, one gets Theorem 4. For a knot diagram D and an α-twisted quandle 2-cocycle ω, the multiset

{ W tw ω (D, C, α) | C ∈ C Q (D) } of α-twisted ω-weights of D coincides with the multiset { W sh ωα (D, C, C sh ) | (C, C sh ) ∈ C sh Q,Z (D), C sh (∆ ex (D)) = 0 }
of its Z-shadow ω α -weights with the exterior region colored by 0.

Proof. According to Lemma 3.3, the first multiset coincides with

{ W sh ωα (D, C, C ind ) | C ∈ C Q (D) }.
But this is precisely our second multiset, since Lemma 3.1 allows to replace any region Z-coloring C sh satisfying C sh (∆ ex (D)) = 0 with C ind . This theorem yields the announced shadow interpretation of twisted quandle cocycle invariants. In particular, combined with Theorem 1, it immediately implies Theorem 3.

Applications

We now turn to applications of our shadow interpretation of twisted quandle cocycle invariants. Some results on shadow invariants are first recalled. Most of them seem to be folklore; their proofs are included for completeness. Their consequences in the twisted setting are then discussed.

Lemma 4.1. The shadow weights associated to a shadow quandle 2-coboundary vanish.

The proof we present is well adapted for higher-dimensional generalizations.

Proof. Take a Q-module M, a map θ : M × Q → A, and a (Q, M)-colored knot diagram (D, C, C sh ). Choose a point p the knot, and examine the value ν(γ) = θ(C sh (∆(γ)), C(γ)) as you travel along the knot, following its direction; here γ is the arc you are on, and ∆(γ) is the region to your right. You pass twice through any crossing , and the sum of the increments of ν(γ) during these two passages is exactly the contribution of to W sh dθ (D, C) (for a positive , it is shown on Figure 5; for a negative one things are similar). Returning to p, you recover the original value ν(γ). Considering such walks along each link component, one concludes that the total increment of ν(γ) -which is precisely In particular, one can talk about the weight W tw

W sh dθ (D, C) -is zero. a b b a ⊳ b m m⊳a m⊳b (m⊳a)⊳b (θ(m ⊳ b, a ⊳ b) -θ(m, a))+ -→ (θ(m, b) -θ(m ⊳ a, b)) = dθ(m, a, b)
[ω] associated to the twisted quandle 2-cohomology class of ω.

Proof. To simplify notations, put d

= d 1 l -d 1 r , d α = d 1 l -αd 1 r .
By the linearity of weights, it suffices to check the triviality of twisted d α θ-weights, with θ : Q → A. An easy direct verification gives (d α θ) α = αd(θ α ) (we use notation (16) and its analogue θ α (m, a) = α -m θ(a)). Lemma 3.3 then implies that

W tw dαθ (D, C) = W sh (dαθ)α (D, C, C ind ) = W sh αdθα (D, C, C ind ) = αW sh dθα (D, C, C ind ),
which vanishes according to Lemma 4.1.

As a second application, we explore an action of a quandle Q on (Q, M)-colorings and its effect on shadow and twisted quandle cocycle invariants.

Lemma 4.3. The set of (Q, M)-colorings of a knot diagram D has the following Q-module structure: the coloring (C, C sh ) ⊳ c is obtained from (C, C sh ) by replacing each employed color x with x ⊳ c. Moreover, for any M-shadow quandle 2-cocycle ω, the ω-weight respects this Q-module structure -that is, for all c ∈ Q, one has

W sh ω (D, C, C sh ) = W sh ω (D, (C, C sh ) ⊳ c). (17)
Proof. We first check that (C, C sh ) ⊳ c is indeed a (Q, M)-coloring. For this, one should show that the coloring rules from Figure 1 remain valid when each color x is replaced with x ⊳ c; this follows from Axiom (1) for Q and for M. This axiom also implies that one actually gets a Q-module structure.

To show relation (17), rewrite the defining property (9) for ω as

ω(m, a, b) -ω(m ⊳ c, a ⊳ c, b ⊳ c) = (ω(m ⊳ a, b, c) -ω(m ⊳ b, a ⊳ b, c)) -(ω(m, b, c) -ω(m, a, c)).
The second line can be interpreted as dω c (m, a, b), where

ω c ∈ C 1 (M, Q, A) is defined by ω c (m, a) = ω(m, a, c), and d = d 1 l -d 1 r .
Consequently, one has

W sh ω (D, C, C sh ) -W sh ω (D, (C, C sh ) ⊳ c) = W sh dωc (D, C, C sh ),
which vanishes according to Lemma 4.1.

This lemma leads to a better understanding of the dependence of shadow invariants from Theorem 1 on the choice of the exterior region color m ∈ M. We will use the notion of orbits of a Q-module M, which are classes for the equivalence relation on M induced by m ∼ m ⊳ a, a ∈ Q.12 Proposition 4.4. For any Q-module M, elements m 1 and m 2 lying in the same orbit of M, and M-shadow quandle 2-cocycle ω of Q, the multisets

{ W sh ω (D, C, C sh ) | (C, C sh ) ∈ C sh Q,M (D), C sh (∆ ex (D)) = m 1 } and { W sh ω (D, C, C sh ) | (C, C sh ) ∈ C sh Q,M (D), C sh (∆ ex (D)) = m 2 }
defining shadow invariants coincide.

Proof. It suffices to check the assertion for m 2 = m 1 ⊳ c. In this case, the map (C, C sh ) → (C, C sh ) ⊳ c establishes a bijection between the (Q, M)-colorings used for defining the two multisets. Lemma 4.3 tells that the ω-weights of the corresponding colorings are the same, so the multisets coincide.

Returning to the twisted setting, one obtains several useful consequences.

Proposition 4.5. An α-twisted quandle cocycle invariant does not change if all its elements are scaled by α.

Proof. The invariants in question are multisets

{ W tw ω (D, C, α) | C ∈ C Q (D) } for α-twisted quandle 2-cocycles ω of Q. Theorem 4 identifies such a multiset with { W sh ωα (D, C, C sh ) | (C, C sh ) ∈ C sh Q,Z (D), C sh (∆ ex (D)) = 0 }.
According to Proposition 4.4, this multiset does not change if the color C sh (∆ ex (D)) of the exterior region of D is imposed to be -1 instead of 0, since the Q-module Z consists of a single orbit. With this new condition, all region colors in C sh should be reduced by 1. Recalling the definition (16) of the shadow quandle 2-cocycle ω α , one sees that this corresponds to multiplying all the shadow ω α -weights by α. Thus such a scaling does not change our invariant.

This proposition can be used to show that certain twisted quandle cocycle invariants are no stronger than usual quandle invariants. For instance, for a finite quandle Q and A = Z[t ±1 ], t-twisted quandle cocycle invariants contain only zeros, since a non-zero weight would imply, by scaling by t, an infinity of pairwise distinct weights, whereas the total number of Q-colorings is finite.

Proposition 4.6. Suppose that for c 1 , . . . , c k ∈ Q and ε 1 , . . . , ε k ∈ {±1}, relation

(• • • (a ⊳ ε 1 c 1 ) ⊳ ε 2 • • • ) ⊳ ε k c k = a (18)
holds for all a ∈ Q; we used notations ⊳ +1 =⊳, ⊳ -1 = ⊳. Then the multiplication by α k i=1 ε i -1 annihilates all α-twisted weights.

Proof. As usual, Lemma 3.3 allows us to work with shadow weights instead of αtwisted weights. A repeated application of Lemma 4.3 gives

W sh ωα (D, C, C ind ) = W sh ωα (D, (• • • ((C, C ind ) ⊳ ε 1 c 1 ) ⊳ ε 2 • • • ) ⊳ ε k c k ). Relation (18) implies (• • • ((C, C ind ) ⊳ ε 1 c 1 ) ⊳ ε 2 • • • ) ⊳ ε k c k = (C, C ind + k i=1 ε i )
, where in the latter region coloring each color is increased by k i=1 ε i . Repeating the argument from the proof of Proposition 4.5, one obtains

W sh ωα (D, C, C ind + k i=1 ε i ) = α -k i=1 ε i W sh ωα (D, C, C ind ). Now, combine everything to get W tw ω (D, C, α) = α -k i=1 ε i W tw ω (D, C, α). For example, if Q contains a "central" element c satisfying a ⊳ c = a for all a ∈ Q, then scaling by α -1 kills all α-twisted weights.
The third application is a shadow enhancement of twisted quandle cocycle invariants. To treat simultaneously the Q-module M we want to color regions with and the Q-module Z (with m ⊳ a = m + 1) used in our shadow interpretation of twisted invariants, we use the following elementary observation:

Lemma 4.7. The direct product M × M ′ of Q-modules M and M ′ is also a Q- module, with the diagonal action (m, m ′ ) ⊳ a = (m ⊳ a, m ′ ⊳ a).
Now, repeating verbatim all the arguments from the previous and the beginning of the current sections for the Q-module M × Z instead of Z, one gets Theorem 5. Take a Q-module M, a fixed m ∈ M, and an α-twisted M-shadow quandle 2-cocycle ω. The multiset of α-twisted M-shadow ω-weights

{ W tw,sh ω (D, C, C sh , α) | (C, C sh ) ∈ C sh Q,M (D), C sh (∆ ex (D)) = m }, where W tw,sh ω (D, C, C sh , α) = a b ε( )α -i( ) ω(C sh (∆( )), a, b),
is a knot invariant 13 . Moreover, cohomologous 2-cocycles yield identical invariants.

The last bonus from our shadow interpretation of twisted quandle cocycle invariants is their higher-dimensional generalization. Indeed, take a Q-colored k -1dimensional knot diagram (D, C) in R k , and an α-twisted quandle k-cocycle ω. Define the α-twisted ω-weight of (D, C) by a formula analogous to (15), using higherdimensional versions of indices and signs for crossings of multiplicity k, and a relevant ordering of adjacent sheets. Lemmas 3.1-3.3 generalize directly. The multiset of α-twisted ω-weights is thus interpreted in terms of higher-dimensional shadow invariants (for more details about the latter, see for instance [START_REF] Fenn | Racks and links in codimension two[END_REF][START_REF] Carter | Surfaces in 4-space[END_REF][START_REF] Przytycki | Cocycle invariants of codimension 2embeddings of manifolds[END_REF]). Moreover, this construction admits a shadow version, similar to that from Theorem 5. Hence any α-twisted (possibly shadow) quandle k-cocycle gives rise to a k -1-dimensional knot invariant.

Positive quandle cocycle invariants via shadows

This section is devoted to the case α = -1. For this α, the twisted quandle cohomology theory is precisely the positive quandle cohomology theory from [START_REF] Cheng | Positive quandle homology and its applications in knot theory[END_REF]. In order to show that positive and -1-twisted quandle cocycle invariants coincide, it remains to compare the weights used in the two constructions: Proof. Is is sufficient to show that for any crossing , the signs ε( )(-1) -i( ) and ε pos ( ) coincide. First, observe that, modulo 2, the coloring C ind reduces to the checkerboard coloring, under the identification 0 = white, 1 = black. Using this identification, the definition of ε pos ( ) (Figure 3) can be restated as on Figure 6 (note that we turned both diagrams so that the over-arc points south-west). Now, if the under-arc points south-east, then ε( ) is 1, and i( ) is the color of the region to the west of ; one checks that the latter satisfies (-1) -i( ) = ε pos ( ), as desired.

A change in the orientation of the under-arc changes ε( ) and i( ), but not ε pos ( ), so equality ε( )(-1) -i( ) = ε pos ( ) is preserved.

{ W sh ω -1 (D, C, C sh ) | (C, C sh ) ∈ C sh Q,Z 2 (D), C sh (∆ ex (D)) = 0 }
of its Z 2 -shadow ω -1 -weights with the exterior region colored by 0.

Note that we replaced our Q-module Z with Z 2 , with the same module operation m ⊳ a = m + 1; this is possible since (-1) m is completely determined by m mod 2.

The theorem interprets positive quandle cocycle invariants as shadow ones. This directly shows their invariance, and gives for free their shadow enhancement, answering a question raised in [START_REF] Cheng | Positive quandle homology and its applications in knot theory[END_REF]. Moreover, this implies analogs of Propositions 4.5 and 4.6 in the positive setting: Proposition 5.2. A positive quandle cocycle invariant is symmetric with respect to 0 (that is, it does not change if all its elements change signs).

Proposition 5.3. Suppose that for some odd k and some c 1 , . . , c k ∈ Q and ε 1 , . . . , ε k ∈ {±1}, relation (18) holds for all a ∈ Q. Then all non-trivial positive weights are of order 2 in the abelian group A.

Twisted quandle cocycle invariants for links

For links with several components, twisted quandle cocycle invariants can be sharpened using a method developed in this section. In what follows, the diagram D represents a link with k ordered components, numbered 1, . . . , k.

First, we need a more subtle notion of index.

Definition 6.1.

• The jth index i j (∆) of a region ∆ in D is the index of ∆ with respect to the jth component of D, in the sense of Definition 2.3.

• The jth index of a crossing is set to be i j ( ) = i j (∆( )). As before, let A be a module over a commutative ring R. Take a collection α = (α O ∈ R * ) O∈Orb(Q) . Definition 6.2.

• An α-twisted quandle 2-cocycle is a map ω : Q × Q → A satisfying (5) and As usual, our shadow interpretation gives for free shadow and higher-dimensional generalizations of the theorem.

Figure 1 :

 1 Figure 1: Quandle and shadow coloringsThese quandle invariants are very powerful, but they do not detect, for instance, the chirality of knots. This flaw was fixed in [CJK + 03] by enriching colorings with weights. Concretely, given an abelian group A, a map ω :Q × Q → A satisfying ω(a, b) + ω(a ⊳ b, c) = ω(a ⊳ c, b ⊳ c) + ω(a, c),(4) ω(a, a) = 0 (5) is called a quandle 2-cocycle of Q. (The word "cocycle" will be justified below.) The ω-weight of a Q-colored knot diagram (D, C) is defined as a sum W ω (D, C) =

Figure 2 :

 2 Figure 2: Quandle cocycle invariants

First, together with

  the arcs of a knot diagram D, one can color its regions (= the connected components of R 2 \D) by elements of a Q-module M. Concretely, define a (shadow) (Q, M)-coloring of D as a Q-coloring C of the arcs of D and an assignment C sh : Regions(D) → M, compatible in the sense of Figure 1 B 8 . The set of such colorings is denoted by C sh Q,M (D). The color of the exterior region ∆ ex (D) of D uniquely determines all the other region colors, hence the (Q, M)-coloring number |C sh Q,M (D)| equals |C Q (D)| • |M| and says nothing new about the knot. The situation changes when weights enter the story. Take an M-shadow quandle 2

  Figure 2.) Theorem 3 ([CES02]). For any α ∈ R * and any α-twisted quandle 2-cocycle ω of Q, the multiset { W tw ω (D, C, α) | C ∈ C Q (D) } yields a knot invariant. This knot invariant is called (α-)twisted quandle cocycle invariant.

Figure 5 :

 5 Figure 5: The total cost of passing through a crossing

Lemma 5. 1 .

 1 For any Q-colored diagram (D, C) and any positive quandle 2-cocycle ω, one has W tw ω (D, C, -1) = W pos ω (D, C).

Figure 6 :

 6 Figure 6: Sign ε pos via region indices 13 For notations, refer to Definition 2.3.

  For a quandle (Q, ⊳), consider the equivalence relation induced by a ∼ a ⊳ b. Corresponding equivalence classes are called orbits, their set is denoted by Orb(Q), and the orbit of an a ∈ Q is denoted by O(a). A Q-coloring C of D assigns elements of the same orbit to all the arcs belonging to the same component of D; for the jth component, this orbit is denoted by C * (j).

Theorem 7 .

 7 α -1 O(c) ω(a ⊳ c, b ⊳ c)ω(a, b)α -1 O(b) ω(a ⊳ b, c) + ω(a, c) + (α -1 O(a) -1)ω(b, c) = 0. (19)• The (α-twisted) ω-weight of a Q-colored link diagram (D, C) is the sumW tw ω (D, C, α) The multiset { W tw ω (D, C, α) | C ∈ C Q (D)} depends only on the underlying link (and not on the choice of its diagram D), and only on the α-twisted quandle cohomology class [ω] of ω.The theorem is proved by interpreting the constructed multisets as shadow invariants. The Q-module one should use here is the following one: Lemma 6.3. The setO∈Orb(Q) Ze O with m ⊳ a = m + e O(a) is a Q-module.Proof. One has to check that e O(b) + e O(c) = e O(c) + e O(b⊳c) , which is obvious since, by definition, b and b ⊳ c belong to the same orbit.

  Remark 6.4. Following[START_REF] Inoue | Quasi-triviality of quandles for link-homotopy[END_REF], the multiset from the theorem can be decomposed into multisets{ W tw ω (D, C, α) | C ∈ C Q (D), C * (1) = O 1 , . . . , C * (k) = O k }, indexed by k-tuples of orbits O 1 , . . . , O k ∈ Orb(Q).The invariance of these smaller multisets follows from the fact that Reidemeister moves and induced local coloring changes preserve the orbit of the colors assigned to the arcs of a given link component.

Operation ⊳ is deduced from ⊳ using (2); we thus often simply speak about a quandle (Q, ⊳).

In this paper by knots we always mean oriented knots and links in R

.

Here and afterwards diagrams with unoriented arcs mean that all coherently oriented versions of these diagrams should be considered.

Q-modules appeared under the name Q-(quandle-)sets ([FRS95]), and are also known as Qshadows ([CN11]).

 5 The notation is slightly abusive but

convenient. 6 Subscripts l & r refer to an interpretation of these maps as left & right differentials, cf. [Leb13].

Note the "double-shadow" situation here.

In our diagrams, region colors are underlined for a better readability.

The term Alexander number is sometimes used instead ([Ale23, CS98, CKS00]).

In our diagrams, a solid crossing stands for a crossing of arbitrary sign.

In other words, it is the region from which all normals to the arcs of point, hence the term.

Orbits can also be thought of as the irreducible components of M . Every Q-module uniquely decomposes as a disjoint union of irreducible Q-modules.

Acknowledgements. The authors are grateful to the organizers of Knots and Low Dimensional Manifolds (a satellite conference of Seoul ICM 2014), where the idea of this paper was born. They would like to thank Zhiyun Cheng for interesting discussions. This work was made possible thanks to JSPS KAKENHI Grants 25•03315, 26287013, and 26400082. V.L. was also supported by a JSPS Postdoctral Fellowship for Foreign Researchers.


Pierre Jounieaux
email: jounieaux@ann.jussieu.fr

Yannick Privat
email: yannick.privat@math.cnrs.fr

Emmanuel Trélat
email: emmanuel.trelat@upmc.fr

Optimal design of

1. Setting of the shape optimization problem 1.1. A brief state of the art.

The literature on the optimal observation or sensor location problems is abundant in engineering applications, where mainly the optimal location of sensors or controllers is investigated. In many contributions, numerical tools are developed to solve a simplified version of the optimal design problem where either the partial differential equation has been replaced with a discrete approximation, or the class of optimal designs is replaced with a compact finite dimensional set (see for example [START_REF] Armaoua | Optimal actuator/sensor placement for linear parabolic PDEs using spatial H 2 norm[END_REF][START_REF] Harris | Optimal sensor location with an application to a packed bed tubular reactor[END_REF][START_REF] Vande Wouwer | An approach to the selection of optimal sensor locations in distributed parameter systems[END_REF] and [START_REF] Morris | Linear-quadratic optimal actuator location[END_REF] where such problems are investigated in a more general setting).

From an engineering point of view, the aim is to optimize the number, the position or the shape of sensors in order to improve the estimation of the state of the system. Fields of applications are very numerous, for example structural acoustics, piezoelectric actuators, or vibration control in mechanical structures just to name a few of them, are interested in such a study.

In this study, we model this issue by maximizing an observability constant with respect to the shape an location of the sensors. We refer to [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF][START_REF] Zuazua | Controllability and observability of partial differential equations: Some results and open problems[END_REF] for a general presentation of observability inequalities for wave like systems. We adopt here the "randomized" approach proposed in [START_REF] Privat | Optimal observability of the one-dimensional wave equation[END_REF][START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF], where the authors introduce a notion of initial data average observability, more relevant for practical issues, and investigate the problem of maximizing this constant with respect to the shape of the sensors, in the case of an internal observation.

Let us mention [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF][START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF], where the authors use a spectral point of view leading to a similar optimal design problem. They investigate the problem of determining the best possible shape and position of the support of a damping term in a one-dimensional wave equation. They provide a complete characterization of the solutions of this problem and highlight the so-called spillover phenomenon (proved rigorously in [START_REF] Privat | Optimal observability of the one-dimensional wave equation[END_REF]).

Boundary observability of the n-dimensional wave equation

Let T be a positive real number and Ω be a bounded open connected set in IR n having a Lipschitz boundary. Consider the homogeneous wave equation with Dirichlet boundary conditions

∂ 2 y ∂t 2 -∆y = 0 (t, x) ∈ (0, T ) × Ω, y(t, x) = 0 (t, x) ∈ (0, T ) × ∂Ω, y(0, x) = y 0 (x), ∂y ∂t (0, x) = y 1 (x) x ∈ Ω. (1)
It is well known that for every (y 0 (•),

y 1 (•)) ∈ H 1 0 (Ω, C) × L 2 (Ω, C
), there exists a unique solution y of (1) satisfying y ∈ C 0 (0, T ;

H 1 0 (Ω, C)) ∩ C 1 (0, T ; L 2 (Ω, C))
. We endow the Lipschitz set ∂Ω with the usual Hausdorff measure H n-1 . The notation H n-1 (Γ) stands for the surface area of Γ.

For a given H n-1 -measurable subset Γ of ∂Ω, consider the observable variable z Γ defined by,

∀(x, t) ∈ ∂Ω × [0, T ], z Γ (t, x) = χ Γ (x) ∂y ∂ν (t, x), (2)
where χ Γ denotes the characteristic function of the set Γ, ν denotes the outward pointing normal vector and ∂y ∂ν is the normal derivative of y.

The system (1)-( 2) is said observable in time T if there exists a positive constant C such that the inequality

C (y 0 , y 1 ) 2 H 1 0 (Ω,C)×L 2 (Ω,C) ≤ T 0 Γ ∂y ∂ν (t, x) 2 dH n-1 dt, (3)
holds for all y 0 (•) ∈ H 1 0 (Ω, C) and y 1 (•) ∈ L 2 (Ω, C). In this case, we denote the so-called observability constant by C T (χ Γ ) the largest constant in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF]. Note that, within the class of C ∞ domains, this observability inequality holds if (Γ, T ) satisfies the Geometric Control Condition (GCC) (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]). An overview of boundary observability results for wave like equations can be found in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF][START_REF] Zuazua | Controllability and observability of partial differential equations: Some results and open problems[END_REF].

Modeling of the problem

For practical purposes in engineering, it is relevant to consider the problem of determining, whenever it exists, the optimal shape and location of sensors. In the sequel, we assume that the surface area of sensors is fixed, equal to LH n-1 (∂Ω), with L ∈ (0, 1).

It could seem natural to maximize the observability constant C T (χ Γ ) over all possible subsets Γ of ∂Ω such that H n-1 (Γ) = LH n-1 (∂Ω). According to (3), the constant C T (χ Γ ) is defined by

C T (χ Γ ) = inf      T 0 Γ ∂y ∂ν (t, x) 2 dH n-1 dt (y 0 , y 1 ) 2 H 1 0 (Ω,C)×L 2 (Ω,C) (y 0 , y 1 ) ∈ H 1 0 (Ω, C) × L 2 (Ω, C) \ {(0, 0)}      . ( 4
)
Such a criterion appears difficult to handle and this modeling is besides not relevant with respect to practical expectations. Indeed, the usual observability constant is deterministic and gives an account for the worst choice of initial data. It is thus pessimistic. In practice, many experiments or measures are carried out. It corresponds in taking into account a huge class of initial data, but not necessarily the whole space H 1 0 (Ω, C) × L 2 (Ω, C) in the definition of the observability constant C T (χ Γ ). Such issues have been widely discussed in a recent series of articles [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF][START_REF] Privat | Optimal shape and location of sensors or controllers for parabolic equations with random initial data[END_REF]. The authors propose a new average observability concept inspired by [START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF] and introduce the notion of randomized observability constant. It can be seen as the mean of the worst L 2 -norm of the observation variable among almost every initial data. We adopt this choice of modeling in the sequel and we describe it now.

Randomized observability constant. Let (φ j ) j∈IN * be a Hilbert basis of L 2 (Ω) consisting of eigenfunctions of the Dirichlet-Laplacian operator on Ω, associated with the negative eigenvalues (-λ j ) j∈IN * .

The solution y of (1) can be expanded as

y(t, x) = +∞ j=1 a j e i √ λj t + b j e -i √ λj t φ j (x), (5)
with

a j = Ω 1 2 y 0 (x) - i λ j y 1 (x) φ j (x) dx, b j = Ω 1 2 y 0 (x) + i λ j y 1 (x) φ j (x) dx (6)
for every j ∈ IN * . Using this expression of y and the energy identity

(y 0 , y 1 ) 2 H 1 0 (Ω,C)×L 2 (Ω,C) = +∞ j=1 λ j (|a j | 2 + |b j | 2 ),
the observability constant C T (χ Γ ) can be written as

C T (χ Γ ) = inf ( aj ),( bj )∈ℓ 2 (R) +∞ j=1 (| aj | 2 +| bj | 2 )=1 T 0 Γ +∞ j=1 a j λ j e i √ λj t + b j λ j e -i √ λj t ∂φ j ∂ν (t, x) 2 dH n-1 dt.
The randomization procedure, introduced in [START_REF] Privat | Optimal observability of the one-dimensional wave equation[END_REF][START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF] consists of replacing C T (χ Γ ) with its randomized counterpart

C T,rand (χ Γ ) = inf ( aj ),( bj )∈ℓ 2 (R) +∞ j=1 (| aj | 2 +| bj | 2 )=1 A T 0 Γ +∞ j=1 β ω 1,j a j e i √ λj t + β ω 2,j b j e -i √ λj t 1 λ j ∂φ j ∂ν (x) 2 dx dt dp ω ,
where (β ω 1,j ) j∈N * and (β ω 2,j ) j∈N * denote two sequences of independent random variables on a probability space (A, P) having mean equal to zero, variance equal to 1 and verifying for all (j, k) in (N * ) 2 , with j = k,

P(β ω 1,j = ±1) = P(β ω 2,j = ±1) = 1 2 and E(β ω 1,j β ω 2,k ) = 0.
The constant C T,rand (χ Γ ) is the largest constant in the so-called randomized observability inequality

C T,rand (χ Γ ) (y 0 ω , y 1 ω ) 2 H 1 0 (Ω)×L 2 (Ω) ≤ E T 0 Γ ∂y ω ∂ν (t, x) 2 dH n-1 dt , (7)
holding for all y 0 (•) ∈ H 1 0 (Ω, C) and y 1 (•) ∈ L 2 (Ω, C), where y ω denotes the solution of the wave equation with the random initial data (y 0 ω , y 1 ω ) determined by their Fourier coefficients a ω j and b ω j deduced from the Fourier coefficients a j and b j of the initial data (y 0 , y 1 ) (see [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF] for the explicit dependence between the Fourier coefficients and the initial datum) by a ω j = β ω 1,j a j and b ω j = β ω 1,j b j . We have the following result. Proposition 1. Let Γ denote a H n-1 -measurable subset of ∂Ω. One has

C T,rand (χ Γ ) = T inf j∈N * 1 λ j ∂Ω χ Γ (x) ∂φ j ∂ν (x) 2 dH n-1 , (8)
where χ Γ denote the characteristic function of Γ.

This leads to consider the following optimal design problem, modeling in our view in a relevant way, the problem of optimizing the placement and shape of sensors along the boundary of Ω.

Optimal design problem. We investigate the problem of maximizing

J(χ Γ ) = inf j∈N * 1 λ j ∂Ω χ Γ (x) ∂φ j ∂ν (x) 2 dH n-1 , (9)
over all possible subsets Γ of ∂Ω of Hausdorff measure H n-1 (Γ) = LH n-1 (∂Ω).

Remark 1. The motivation for maximizing the constant C T,rand (χ Γ ) instead of the observability constant C T (χ Γ ) has been discussed previously. The issue of maximizing the functional C T (χ Γ ), even if it is not so relevant for practical applications, remains plausible and is intrinsically interesting. Nevertheless, it appears to be very difficult from the theoretical and numerical points of view, due to the presence of crossed terms Γ ∂φj ∂ν (x) ∂φ k ∂ν (x)dH n-1 measuring the interaction over Γ between distinct eigenfunctions.

2. Study of the optimal design problem (9)

Investigating the optimal value

Define the admissible set

U L = χ Γ | Γ is a H n-1 -measurable subset of ∂Ω of measure H n-1 (Γ) = LH n-1 (∂Ω) . (10)
Note that the problem ( 9) is sup

χΓ∈U L J(χ Γ ).
Since the set U L is not compact for the L ∞ weak-star topology, it is usual to introduce a convexified version of the problem [START_REF] Ho | Observabilité frontière de l'équation des ondes[END_REF]. For this reason, we consider the convex closure of the set U L ,

U L = a ∈ L ∞ (∂Ω; [0, 1]) | ∂Ω a(x) dH n-1 = LH n-1 (∂Ω) , (11)
and the relaxed formulation of the problem (9),

sup a∈U L J(a), (12)
where

J(a) = inf j∈N * 1 λ j ∂Ω a(x) ∂φ j ∂ν (x) 2 dH n-1 . ( 13
)
Proposition 2. Let L ∈ (0, 1). The relaxed problem (12) has at least one solution a * ∈ U L .

In the following results, we may use the following assumptions on the basis (φ j ) j∈IN * of eigenfunctions under consideration.

Uniform boundedness. There exists A > 0 such that for every j

∈ IN * ∂φ j ∂ν L ∞ (∂Ω) ≤ Aλ j . ( 14
)
Remark 2. Assumption ( 14) holds true for whenever Ω is a square or a disk (see Section 2.2).

Weak Quantum Ergodicity of Boundary values (WQEB) property. There exist a subsequence of dH n-1 is converging vaguely to the uniform measure 2 n|Ω| dH n-1 .

Remark 4. We insist on the fact that, even if WQEB is satisfied for smooth ergodic domains, very few of them may satisfy the more restrictive SQEB property. For instance in dimension 2 the disk is an example of domain Ω which satisfies this last property. At the opposite, SQEB does not hold true for the square (however this particular case, studied in section 2.2, satisfies the WQEB property) or general rectangles.

It is clear that, under WQEB and ( 14) there exists a subsequence of 1 λj ∂φj ∂ν 2 converging to 2 n|Ω| for the L ∞ (∂Ω) weak-star topology, then one has J(a) ≤ 2 LH n-1 (∂Ω) n|Ω| , for every a ∈ U L . For every x 0 ∈ Ω, define

a x0 (x) = LH n-1 (∂Ω) n|Ω| x -x 0 , ν ,
for every x ∈ ∂Ω, where •, • denotes the standard Euclidean inner product in IR n . Using the instrumental Rellich identity (see for example [21, chap.7, p.232])

2λ j = ∂Ω x -x 0 , ν ∂φ j ∂ν (x) 2 dH n-1 , (15)
we infer the following result.

Proposition 3.

If Ω satisfies WQEB and (14), and if there exists x 0 ∈ Ω such that 0 ≤ a x0 (x) ≤ 1 for every x ∈ ∂Ω, then a x0 is a solution of the relaxed problem [START_REF] Jounieaux | Optimal boundary observation domain for the wave equation[END_REF], and

max a∈U L J(a) = 2 LH n-1 (∂Ω) n|Ω| .
In the next remark, we provide a class of examples where the existence of x 0 such that a x0 ∈ U L is true. 1 Actually in these references it is proved that a density one subsequence of 1 λ j ∂φ j ∂ν 2 dH n-1 converges to the uniform measure under ergodic assumptions.

Remark 5. Set x 0 = 0. Assume that Ω is convex, and that there exist two balls B e and B i of IR n , centered at the origin, of radii R e and R i such that

B i ⊂ Ω ⊂ B e and R e R i n ≤ n 2 4L .
Then, a 0 belongs to U L . Indeed, it follows from the convexity of Ω, that 0 ≤ x, ν ≤ R e for every x ∈ ∂Ω. Recall that in the class of convex sets, the volume and the perimeter are monotone increasing for the inclusion. Thus, |B i | ≤ |Ω| and

H n-1 (∂Ω) ≤ H n-1 (∂B e ), so that a 0 ≤ 4L n 2
Re Ri n ≤ 1 on ∂Ω. for every x ∈ Ω. Some admissible choices of domains Ω in dimension 2 for which a 0 ∈ U L are drawn in Figure 1. And for some admissible choices of domains Ω, Figure 2 gives the values of a 0 .

x 0 ∈ Ω such that 0 ≤ a x0 (x) ≤ 1 for every x ∈ ∂Ω, then sup χΓ∈U L inf j∈N * 1 λ j ∂Ω χ Γ (x) ∂φ j ∂ν (x) 2 dH n-1 = sup a∈U L inf j∈N * 1 λ j ∂Ω a(x) ∂φ j ∂ν (x) 2 dH n-1 . (16)
In other words there is no gap between the problem (9) and its convexified version [START_REF] Jounieaux | Optimal boundary observation domain for the wave equation[END_REF].

Corollary 1. Under the assumptions of Theorem 1, we have the optimal value for the problem (9)

sup χΓ∈U L J(χ Γ ) = 2 LH n-1 (∂Ω) n|Ω| .
Now the question is to know if this supremum is reached on U L or not. In this view we propose in the next section the study of some particular cases.

Solving (9) in some particular cases

In this section, we investigate some particular cases for which SQEB may not hold true (cf remark 4) but however the conclusion holds, highlighting that SQEB is not sharp in Theorem 1.

Proposition 4. Let L ∈ [0, 1].
• Assume that Ω = (0, π) 2 . we have,

sup χΓ∈U L J(χ Γ ) = max a∈U L J(a) = J(L) = πL.
Moreover,the supremum over U L is reached if and only if 0, 1 4 , 1 2 , 3 4 , 1 . • Assume that Ω = B(0, R). We have,

sup χΓ∈U L J(χ Γ ) = max a∈U L J(a) = J(L) = πRL.
Moreover, the supremum over U L is reached if and only if L ∈ {0, 1 4 , 1 2 , 3 4 , 1}. We are actually able to characterize explicitly all optimal domains whenever they exist (see [START_REF] Jounieaux | Optimal boundary observation domain for the wave equation[END_REF]). On figures 3 and 4 we provide the drawings of several of them.

Truncated problem and numerical simulations

While investigating the optimal value of J on U L , we had to make the two main assumptions SQEB and UB. A natural question now is how to approximate the optimal value when those assumptions are not satisfied. In this view we propose here to truncate the criterion (13) at some order N ∈ IN * , and thus to consider the problem sup

a∈U L J N (a) (17)
where

J N (a) = inf 1≤j≤N 1 λ j ∂Ω a(x) ∂φ j ∂ν (x) 2 dH n-1 (18)
Moreover the study of the truncated problem finds some interest from an engineering point of view, considering that it may be more relevant to take into account only a finite number of modes in this framework. The following Γ-convergence result states that, when there exists a solution of Problem [START_REF] Privat | The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF] in U L for every N ∈ IN, then the optimal values of (18) converge to the optimal value of Problem [START_REF] Jounieaux | Optimal boundary observation domain for the wave equation[END_REF]. Note that, when the assumptions of Theorem 1 are satisfied, it yields an estimate of the optimal value for the initial problem (9). Proposition 5. J N Γ-converges to J in the sense of De Giorgi. Moreover, when there exist a sequence (χ * Γ N ) N ∈IN of maximizers of J N , then χ * Γ N converges (up to a subsequence) for the L ∞ weak-star topology to a maximizer of J in U L and

max a∈U L J(a) = lim N →+∞ J N (χ * Γ N ).
Remark 6.

(1) The sequence (χ * Γ N ) N ∈IN * is however not in general a maximizing sequence of J. Indeed, using a strategy close to the one used in [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] to emphasize the spillover phenomenon, it is possible to show in the case of the square Ω = (0, π) 2 that for ε > 0 small enough, there exists N 0 ∈ IN * such that

J(χ * Γ N ) < max χΓ∈U L J(χ Γ ) -ε = L 2 -ε,
for every N ≥ N 0 . (2) For some particular cases (for instance the square and the ellipse), we have the existence and the uniqueness of a solution for Problem [START_REF] Privat | The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF] in U L . In the following paragraph we propose some numerical computation of optimal domains for the truncated problem [START_REF] Privat | The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF] for those particular cases. Moreover, in an ongoing work [START_REF] Jounieaux | Optimal boundary observation domain for the wave equation[END_REF] we prove existence and uniqueness of a solution of Problem [START_REF] Privat | The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF] under generic assumptions on Ω. This result comes from the fact that for analytic domains, if the functions ∂φj ∂ν 2 are independent, one has that a solution must be a level-set of a combination of those functions. This independance holds true under generic assumptions. Anyway, the truncated problem [START_REF] Privat | The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF] has solutions in U L , for several particular domains such as the square, the disk and the ellipses, computed hereafter.

The following simulations were obtained with a direct method consisting in discretizing the underlying differential equations, the control, and to reduce the optimal control problem to some minimization problem with constraints. We used the code IPOPT (see [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]) combined with AMPL (see [START_REF] Fourer | AMPL: A modeling language for mathematical programming[END_REF]) on a standard desktop machine. Figures 5 and6, concern the case of the square and the ellipse. The case of the disk computed in Figure 7, is slightly different. In this case we do not have the unicity of a solution in U L . Nevertheless in this case one can find easily which are the optimal domains for Problem [START_REF] Privat | The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF]. Notice that we do not have unicity of the solution and thus propose different examples of solutions.

Figure 1 .Figure 2 .

 12 Figure 1. Two domains Ω in dimension 2 for which a 0 ∈ U L

4 Figure 3 . 4 Figure 4 .

 4344 Figure 3. Ω = (0, π) 2 . Some particular solutions for L = 1/4, L = 1/2 and L = 3/4.

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. The square for 2 and 5 modes (L = 0.3, and L = 0.7)


Gérald Portemont
email: gerald.portemont@onera.fr

Cécile Davoine

Bertrand Langrand

Vincent Marcadon

Cellular materials made of stacked tubes : influence of the manufacturing process on the dynamic behaviour of the constitutive material Part II : mechanical behaviour

influence of the manufacturing process on the dynamic behaviour of the constitutive material Part II : mechanical behaviour.

Matériaux cellulaire par empilement de tubes: influence du procédé de fabrication sur le comportement dynamique du matériau constitutif. Partie II : comportement mécanique. par

Introduction

Cellular materials are very promising as lightweight aeronautical frames thanks to their superior specific mechanical properties such as impact resistance [START_REF] Evans | Multifunctionality of cellular metal systems[END_REF]. However, because of the processing routes and heat treatments used in the production process, the material within the cell walls may behave differently from the bulk, and therefore the in situ mechanical properties are often unknown [START_REF]Effects of High Strain Rate on Properties and Microstructure Evolution of TWIP Steel Subjected to Impact Loading[END_REF][START_REF] Wang | Mater. Effects of grain boundary carbides on the mechanical properties of Inconel 600[END_REF]. The present works aims at investigating the link that exists between the processing of cellular architectures and the mechanical properties of their constitutive material. The cellular material studied is an Inconel 600® tube stacking brazed using a nickel-phosphorus alloy (Figure 1).

(a) (b) Figure 1: Cellular structures considered: (a) square stacking core, (b) hexagonal stacking core.

The experimental works have been performed in order to analyse the mechanical characterisation of regular tube stackings that focuses on the evolutions of both microstructures and mechanical properties of Inconel 600® resulting from brazing heat treatment and diffusion of the brazing compounds [START_REF] Marcadon | Mechanical behaviour of hollow-tube stackings: Experimental characterization and modelling of the role of their constitutive material behaviour[END_REF]. The confrontation of the mechanical properties and microstructures using electron back-scattered diffraction analyses (EBSD) are presented in more in a paper by Davoine et al. [START_REF] Davoine | Mechanical behaviour of hollow-tube stackings: Cellular materials made of stacked tubes: influence of the manufacturing process on the dynamic behavior of the constitutive material. Part I: microstructure[END_REF]. Different material configurations are thus characterised and compared, involving both tensile tests from quasi-static to dynamic loads and. In order to discuss the influence of the manufacturing process of tube stackings on the mechanical properties of their constitutive material, it has been proposed to perform uni-axial tensile tests at various strain rates on tubular specimens for three different materials configurations. The first one was made of base metal material (Inco600BM), the second one was affected by the brazing heat treatment (Inco600HT), and the latest was submitted to the NiP layer deposition then followed by the brazing heat treatment (Inco600NiP).

Experimental device for the material behaviour characterisation under high strain rate

2.1

Experimental set-up In order to discuss the influence of the manufacturing process of tube stackings on the mechanical properties of their constitutive material an Inconel 600® here, it has been proposed to perform uniaxial tensile tests at various strain rates on tubular specimens for the three different material configurations. The cellular materials in Figure 1 being possibly subjected to dynamic loads (i.e. fan blade off application for instance), the mechanical behaviour of the three tube material configurations has also been studied under different strain rates in order to analyse possible viscous effects.

The strain rate dependency of the three various Inconel600® configurations was studied with a uniaxial tensile device developed for tubular specimens. The device in Figure 2 makes it possible to clamp the tubular specimen all around its section. A rigid pin was inserted into the tubular specimen in each gripping area (upper and lower holders) of the specimen in order to avoid its deformation during the tightening before testing. This specific device has been developed especially for high strain rate testing by using a hydraulic jack. Each specimen was subjected to a uni-axial tensile loading at a given strain rate, by using a hydraulic jack and the experimental device (Fig. 2).

Force, displacements and strains were measured the same way for the tests performed in quasi-static and dynamic loading conditions. The force was measured by a piezo-electric load cell and the elongation was measured by an optical extensometer and the black/white mark on one side of the specimen (Fig. 3). A strain gauge was bonded on the other side of each specimen as presented in Figure 3. Tests were performed at various imposed velocity varying between 6mm.min -1 (1.10 -3 s -1 ) and 2m.s -1 (100s -1 ). The actual strain rates were analysed for each test using the strain measurement given by the strain gauge. For each material configuration, a set of three tests to the minimum was performed at the same displacement rate to assess the experimental or material discrepancy.

Design of the tubular specimen

Using smooth specimens, plastic strain developed near the gripping areas for very low levels of average strain. The analysis of the material behaviour was consequently limited to engineering strain levels lower than 5% using the smooth tubular specimen. The machined tubular specimen shown in Figure 3 has been especially developed to avoid plastic strain localisation and failure near the gripping areas and to make the analysis of the tube material behaviour possible under a wide range of average strains and strain rates. Tubular samples were machined from 75mm length tubes with external and internal diameters equals to 6mm and 4mm respectively. The tubes were originally produced using extrusion and cold drawing. The external diameter in the centre of the specimen corresponds to the external diameter of the tubes implemented in the cellular materials in Figure 3, i.e. 5mm. This is particularly important that the tubular specimens for tensile testing and the tubes within the cellular materials have a similar diffusion and penetration of the brazing metal (i.e. the NiP alloy) into the Inconel 600® base material. Hence the set of Inco600NiP specimens has been coated by a 15µm thickness NiP layer. This thickness has been calibrated in order to produce a similar diffusion and penetration of the brazing metal into the base metal, compared to the actual manufacturing process. Preliminary tests have been performed with a stereo digital image correlation (DIC) system to attest the strains uniformity over the gauge length of the machined tubular specimen (Fig. 4). These tests results were compared with the others tests performed on the smooth tubular specimens with external and internal diameters that equalled to 5mm and 4mm respectively. The stress versus strain response obtained for both specimens (smooth and machined specimens) was very close until the plastic localisation in the smooth specimen.

Experimental results

The strains measured by both the strain gauge and the optical extensometer delivered the same measurements until the failure of the strain gauge. The stress time history shows high frequency oscillations for the tests performed at a displacement rate of 1m.s -1 . These oscillations came from the first natural frequency of the experimental device (i.e., lower holder in Figure 2) that disturbed the measurements of the load cell during the dynamic test. Performing tests at displacement rates higher than 1m.s -1 would have increased the amplitude of these oscillations and digital filtering would have been necessary to analyse the results. That is reason why the dynamic tests were limited to displacement rates of 1m.s -1 in our study. Experimental results presented in the paper are consequently raw data. These measurements are analysed to define the engineering stress versus engineering strain diagrams obtained at different strain rates.

Experimental results show similar standard deviations whatever the displacement rate applied to the specimen, except for the tests performed at 0.2m.s -1 . This can be explained by the fact that the hydraulic jack used to perform the tests could not maintain a constant strain rate at this specific displacement rate. However, the standard deviation is about 7% and remains acceptable for dynamic tests.

Inco600BM: as-received tube material

The material properties evolve on the strain rate. Lee et al. [6] and Wang et al. [7] have already observed strain rate dependency for other Inconel® materials. Whereas Young's modulus, E is not dependent on the strain, both the yield and ultimate stresses, σ y and σ u , increase significantly and gradually with the strain rate. Results show that the material behaviour of the Inco600BM base material is elasto-visco-plastic. The rate effect appears from the tests performed at 0.7s -1 . Complementary tensile tests have been performed at displacement rates varying between 1.10 -4 m.s -1 and 8.10 -3 m.s -1 in order to better estimate the strain rate threshold above which a viscous effect appears. Results were not influenced by the strain rate until values close to 0.28s -1 .

The engineering stress versus engineering strain diagrams obtained at different strain rates are presented in Figure 5 (featuring the test at 0.28s -1 ). The engineering strain is those delivered by the strain gauge until the failure of the sensor. These diagrams will be used in the following section to analyse the true stress versus true plastic strain relationship observed at different strain rates and then identify the parameters of a constitutive material law.

Figure 5: Engineering stress vs. strain diagram for the Inco600BM material. The engineering strain is given by the strain gauges

Inco600HT: tube material affected by the heat treatment

This section deals with the experimental characterisation of Inconel®600 that has been subjected to the heat treatment only before tensile testing. The heat treatment was applied once the specimens were machined. The observation of the specimen shows no permanent deformation after the heat treatment. The tubular specimens were tested at various displacement rates using the same aforementioned experimental protocol. The standard deviation increases a little in dynamic loading conditions, but experimental results remain satisfactory (e.g. 2.4% for the yield stress in static loading conditions and 7.6% at a displacement rate of 1 m.s -1 ). They do not highlight strain rate effects on the yield and ultimate stresses when Inconel ®600 is subjected to the heat treatment only. The engineering stress versus engineering strain diagrams obtained for the Inco600HT material at different strain rates are presented in Figure 6. Once again, the engineering strain is that delivered by the strain gauge until the failure of the sensor. Figure 6: Engineering stress vs. strain diagram for the Inco600HT material. The engineering strain is given by the strain gauges

Inco600NiP: tube material after the NiP layer deposition and the heat treatment

This section addresses the experimental characterisation of the Inconel R 600 material that was covered with a NiP layer and then subjected to the heat treatment before tensile testing. The heat treatment was exactly the same as the one already described in Section 2.2. The tubular specimens were tested by using the same experimental protocol, as previously. For this set of experiments, the strain gauges debonded for very small strain level (< 3%) even when using epoxy adhesives, such M-Bond AE-10 by Vishay, and cure cycles for polymerisation. The NiP layer turned to be brittle making failure of the strain gauges too. Consequently, it has been decided to consider only the optical extensometer for the strain measurement.

Results exhibit a larger discrepancy for this particular material configuration. It can be explained by the presence of the 50µm thickness NiP layer which was applied to the external and internal surfaces of the tube. When the heat treatment was operated, the final thickness of the NiP layer over the specimen gauge length could not be totally controlled. If the external diameter could be measured in the gauge length of the specimen, the actual internal diameter could not, making uncertainties regarding the cross section of the specimen. Consequently, material behaviour differences observed at the different strain rates could come from differences between the actual section of each specimen and its estimate using the only external diameter. Other tensile tests have been performed at higher strain rates and the results observed at 200s -1 remain similar to those obtained at 20s -1 as shown in Figure 7. If the material behaviour is affected by the strain rate, the rate effect is below the experimental discrepancy.

Discussion

The tensile behaviours observed under quasi-static loading are compared for the three material configurations considered. The engineering strain given by the optical extensometer is used to compare the mechanical responses until failure (Figure 8). Whereas the heat treatment significantly decreases the initial yield stress of the base material, the diffusion and the penetration of phosphorus into the tube material does not increase the initial yield stress of the metal. Yield stresses measured for both Inco600HT and Inco600NiP materials are very similar. However, the diffusion and the penetration of phosphorus results in an increase of the material hardening. Inco600BM tubes were obtained by cold stretching that induced plastic strain and increased the material yield stress. Thus the heat treatment acts such as a recovery of the microstructure and the mechanical properties of the tube material after its mechanical processing (e.g., cold drawing). When offsetting the total strain of the Inco600BM material to project the yield stress of the Inco600BM material on the stress versus strain diagram of the Inco600HT material (here a strain offset of about 0.03 is added to the total strain), it is observed that the hardening of both material configurations (Inco600BM and Inco600HT) are very similar up to strain levels close to 0.2.It is worth noting that the strain to fracture of the Inco600HT material is approximately the same than that of the Inco600BM material with the aforementioned strain offset.

Conclusions

The present works aims at investigating the link that exists between the processing of cellular architectures and the mechanical properties of their constitutive material. The cellular material studied is an Inconel 600® tube stacking brazed using a nickel-phosphorus alloy. The experimental works have been conducted in order to analyse the mechanical characterisation of regular tube stackings that focuses on the evolutions of mechanical properties of Inconel 600® resulting from brazing heat treatment and diffusion of the brazing compounds.

Different material configurations are thus characterised and compared, involving both tensile tests from quasi-static to dynamic loads. In order to discuss the influence of the manufacturing process of tube stackings on the mechanical properties of their constitutive material, it has been proposed to perform uni-axial tensile tests at various strain rates on tubular specimens for three different materials configurations. The first material configuration tested is the tube material base metal (Inco600BM), the second material configuration is related to the tube material affected by the brazing heat treatment only (Inco600HT) and the latest material configuration is related to the Inconel 600 tube material submitted to the NiP layer deposition then followed by the brazing heat treatment (Inco600NiP).

A specific device has been developed especially for high strain rate testing by using a hydraulic jack. Results show that a viscous effect on the mechanical behaviour of Inconel 600 exists in the tubes before heat treatment, the material behaviour is elasto-visco-plastic. However, the heat treatment acts such as a recovery treatment on the plastic properties of the tube material that eliminates viscous effects. Different mechanical properties have been identified according to Johnson-Cook's model [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF].

Electron back-scattered diffraction (EBSD) post-mortem analyses of the tensile specimens show a strong texture too, resulting from the tensile loading and a quite homogeneous intragranular misorientation is observed inside the grains, reflecting large plastic deformations. EBSD analyses are presented in more details in a paper by Davoine et al. [START_REF] Davoine | Mechanical behaviour of hollow-tube stackings: Cellular materials made of stacked tubes: influence of the manufacturing process on the dynamic behavior of the constitutive material. Part I: microstructure[END_REF].

The experimental investigation has been proposed to characterize of the tube stackings in Figure 1 for the dynamic compressive loading. All experimental results (tensile tests on machined tubular specimen, compressive tests on the tube stackings) will be used in future works addressing the finiteelement modelling of the mechanical behaviour of tube stackings.

Figure 2 :

 2 Figure 2: Experimental set-up for the tensile tests at high strain rates on tubular specimens

Figure 3 :

 3 Figure 3: Tubular specimen for tensile testing. The strain was measured on one side of the specimen by an optical extensometer with black and white marks, and on the other side by a strain gauge.

Figure 4 :

 4 Figure 4: Tensile strain distribution measured by stereo Digital Image Correlation for different uniaxial tensile stresses (machined tubular specimen)

Figure 7 :

 7 Figure 7: Engineering stress vs. strain diagram for the Inco600NiP material. The engineering strain is given by the strain gauges

Figure 8 :

 8 Figure 8: Engineering stress vs. strain diagram for the three material configurations tested for 1.10 -3 s -1


D Delhaye
email: david.delhaye@onera.fr

F.-X Ouf

D Ferry

D Gaffie

O Penanhoat

S Peillon

The MERMOSE project: characterization of particulates emissions of a commercial aircraft engine: from morphology to chemical composition

Keywords: Aircraft engine, combustion aerosol, size, composition, morphology

Air transport commercial activities increase continuously trough the years [START_REF] Airbus | International Standards and Recommended practices, annex 16, volume II Aircraft engine emissions[END_REF]Boeing, 2013). Their environmental impacts on air quality and global warming [START_REF] Lee | Transport impacts on atmosphere and climate: Aviation[END_REF] are a main concern and still request nowadays some investigation.

In this framework, the French government funded some research projects in order to better understand the process that lead to the formation of condensation trails, to prevent and manage them. The MERMOSE project (http://sites.onera.fr/MERMOSE/) is the first brick to fill this gap. It aims to provide modern aircraft engine emission data and knowledge on emitted particles reactivity mainly with water.

During this project a field campaign has been performed behind a complete SaM 146 aircraft engine at Snecma test bench. Together, ONERA, IRSN, Snecma and CNRS (CINaM, PhLAM) made gas and particles measurements with a single hole probe that allowed sampling at 5 cm behind the engine exit. Cartographies through 16 points at various engine thrust has been done, from 30% engine maximum take-off thrust to 100%.

The experimental set-up has been composed of 4 lines, all connected to the same probe. The first one was dedicated to gas measurements following the certification guidelines (ICAO, 1993). One line dealt with sampling for laboratory characterizations, microscopy (TEM) and chemical analysis (EC/OC ratio, XREDS, FTIR, laser and ionic desorption mass spectrometry, Raman spectroscopy). Taking into account the SAE E-31 recommendations, two lines have been built to make on-line monitoring of size distributions (SMPS+C, SMPS+E, DMS 500), mass (MAAP, Pegasor, filters) and number (CPC, Pegasor) concentrations and particle surface area density (NSAM).

The modal diameter of the particles is ranging from 17 nm to 55 nm (increasing with the percentage of the maximum thrust), the particles denote a fractal morphology and are composed of nanoparticles of nearly 15 nm in diameter. They are mainly composed of carbon (with traces of O, S and Ca) and their EC/OC ratio increases as a function of the maximum thrust from 20/80 for 30% to 87/13 for 100%. This is confirmed by fine surface chemical composition mass spectrometry analysis performed using laser (L2MS) and ionic (SIMS) desorption. The maximum number of particles emitted has been detected for a 85% engine maximum thrust. Cartographies exhibit homogeneous size particle emission though engine plan exit. Some variations are observed for mass, number and surface area densities.

Figure 1 .

 1 Figure 1. SMPS size distributions for various engine maximum thrusts at two different locations behind a SaM1 146 aircraft engine.

Acknowledgements

This work was supported by the MERMOSE project for the characterization of emissions by aircraft engines and sponsored by DGAC (French national funds).