Marc Soret

Marina Ville

Random walks and the symplectic representation of the braid groups

We consider the symplectic representation ρ n of a braid group

If n is odd, we derive that the n-braids β verifying |∆ β (-1)| ≤ C for some constant C form a transient set: here ∆ β denotes the Alexander polynomial of the closure of β.

We also derive that for a random 3-braid, the quasipositive links (βσ i β -1 σ j ) p have zero signature for every integer p and 1 ≤ i, j ≤ 2.

As an example of such braids, we investigate the signature of the Lissajous toric knots with 3 strands.

Preliminaries 1.Representations of the braid group

If B(n) is the group of braids with n strands, we denote by B its reduced Burau representation which represents braids as (n -1) × (n -1) matrices with values in Z[t, t -1 ]. The specialization of B for t = -1, denoted B -1 , is called the integral reduced Burau representation. The representation B -1 is closely related to the symplectic representation of B(n) which we now recall, using the description of A'Campo ([A 'C]). The braid group B(n+1) acts on the homology H 1 (X, Z) of a surface X such that H 1 (X, Z) is freely generated by n loops. This action is conjugate to the 1 integral reduced Burau representation. Concretely speaking, this means the existence of a basis of H 1 (X, Z) w.r.t. which the matrices representing the elements of B(n + 1) are the ones given by B -1 . This action of B(n + 1) preserves the intersection form I on H 1 (X, Z).

1. If n is even, n = 2l, the form I is symplectic nondegenerate on H 1 (X, Z).

2. If n is odd, n = 2l + 1, the kernel of I is a line K; B -1 restricts to the identity on K and acts symplectically on the quotient H 1 (X, Z)/K Thus, in both cases, we get a symplectic representation

ρ n : B(n + 1) -→ Sp(2l, Z) (1)

Random walks on groups

If µ 1 , µ 2 are probability measures on a discrete group G with finite support, their convolution is defined for g ∈ G as

µ 1 ⋆ µ 2 (g) = h∈G µ 1 (g -1 h)µ 2 (h) (2)
Definition 1. A probability µ on a discrete group G is nondegenerate if supp(µ) generates G as a semigroup.

Malyutin ( [Ma]) defines the right random µ-walk as the Markov chain which starts at the identity of G and with transition probability P (g, h) = µ(gh -1 ). If X is a subset of G, the probability that this walk hits X at the k-th step is µ ⋆k (X).

The signature of a link

The signature is a classical invariant of links and knots; (see for example [Mu]). If γ is a link on S 3 which bounds a Seifert surface Σ in S 3 , we consider the intersection form

I : H 1 (Σ) × H 1 (Σ) -→ Z (α, β) → lk( α, β)
where lk is the linking number and α is the link obtained by pushing α in the direction normal to Σ in S 3 . Given a base for H 1 (Σ), we define the matrix V of I: the signature of γ is the signature of the bilinear form defined by the matrix V + t V . We point out that the literature contains two different definitions of the signature which coincide up to sign. The signature of a link can be computed on a braid representing it and Gambodau-Ghys have estimated the signature of a product of two braids in terms of the signatures of the two braids ([G-G]). We recall (cf. [Mu]) that the for a knot K,

|sign(K)| ≤ 2g 4 (K) (3)
where g 4 denotes the topological 4-genus. Thus a slice knot has zero signature but the converse is not true and we will see more examples of that here.

Positive links and quasipositive braids

A positive link is a link which has a diagram with all positive crossings.

Theorem 1. ( [Pr]) A positive link has strictly negative signature.

So it is interesting to look at the signature of quasipositive braids:

Definition 2. ([Ru]) A braid β is quasipositive if it is a product of conjugates of positive braid generators β = k i=1 γ j i σ j i γ -1 j i (4)
If we close the quasipositive N-braid (4) in a link β, Rudolph proved ( [Ru]) that

χ 4 ( β) = N -k (5)
where χ 4 denotes the largest Euler characteristic of a smooth surface in B 4 bounded by β.

Tanaka ( [Ta]) constructed examples of quasipositive braids with zero signature.

Transient sets in the braid groups from quasimorphisms

We recall that a quasimorphism on a group G is a function Φ : G -→ R such that there exists a positive constant C with

∀a, b ∈ G, |Φ(ab) -Φ(a) -Φ(b)| ≤ D Φ (6)
where D Φ is called the defect of Ψ. Quasimorphisms give us exemples of transient sets in the braid groups:

Theorem 2. ( [Ma]) Let G be a countable group and µ a nondegerate probability on G (see Definition 1). If a subset S of G has bounded image under an unbounded quasimorphism Φ, then the probability that the right random µ-walk hits S at the k-th step tends to zero as k tends to infinity.

We will see below how Gambodau-Ghys's formula for the signature turns it into a quasimorphism. Several other link invariants such as χ 4 can be used to define quasimorphisms on the braid group ( [Br], [B-K]). Other exemples of transient sets on the braid groups have been constructed by Ito [It] 2 The results Theorem 3. Let P be a polynomial in (2l) 2 variables with coefficients in Z. If M = (m ij ) ∈ Sp(2l, Z), we let P (M) = P (m 11 , ..., m 12l , m 21 , ..., m 22l , m 2l1 , ..., m 2l2l ). Suppose that P does not vanish identically on Sp(2l, Z). Then the set

{β ∈ B(n)/P (ρ n (β)) = 0}
is transient for the right random µ-walk.

The Alexander polynomial

Corollary 1. Let C be a real number. If 2s+1 is an odd integer, the following set is transient for the right random µ-walk on B(2s + 1)

{β ∈ B(2s + 1)/|∆ β (-1)| < C} (7)
where ∆ β denotes the Alexander polynomial of the closure of β.

Proof. The Alexander polynomial can be expressed in terms of the Burau representation B ( [Bu]),

∆ β (t) = 1 -t 1 -t n det(B -t (β) -I) (8) If n is odd, ∆ β (-1) = det(ρ n (β) -I)
and the set ( 7) is a finite union of sets described in Theorem 3.

REMARK. If n is even, 1 -(-1) n = 0 and det(B -1 (β) -I) = 0 since B -t acts on K as the identity; thus the formula ( 8) is not easy to use. Nevertheless, it seems likely that Theorem 1 is true also for even n's.

The signature of 3-braids

Corollary 2. Let µ be a nondegenerate probability on the braid group B(3). For the right random µ-walks on B(3), the set

S = {β ∈ B(3)/∃n ∈ N, ∃i, j ∈ {1, 2} such that sign (βσ i β -1 σ j ) n = 0} (9) is transient.
By contrast, (5) tells us that for every n, χ 4 (βσ

1 β -1 σ 1 ) n = 3 -2n; in particular, if (βσ 1 β -1 σ 1 ) n closes into a knot, g 4 ((βσ 1 β -1 σ 1 ) n ) = n -1.
In §5, we recall the Lissajous toric knots with 3 strands (cf. [S-V]); they are naturally of the form (βσ 2 β -1 σ ±1 1 ) n and we prove Theorem 4. A Lissajous toric knot K(3, p, q) has zero signature unless it is isotopic to a torus knot.

We conclude by statistical estimates, done with SAGE.

These results suggest possibles generalizations.

Question 1. Let n ∈ N and let I be a numerical link invariant which is unbounded on the closures of the links represented by a n-braid. If C is a constant, is the set {I( β) ≤ C} transient for the right random µ-walks?

Question 2. For an integer m, we set

A (1) m = 1≤2i+1≤m σ 2i+1 A (2) m = 1≤2i≤m σ 2i
Is the following set S m transient in B(m) for the right random µ-walks?

S m = {β ∈ B(m)/∃n ∈ N, ∃i, j ∈ {1, 2} such that sign (βA (i) m β -1 A (j) m ) n = 0}
3 Random walks: proof of Theorem 3

Similarly to Rivin in [Ri], we introduce the finite groups Sp(2l, Z p )'s for a prime number p > 2.

A theorem of A'Campo ([A 'C], Theorem 1 (1)) shows that the representation B(n) -→ Sp(2l, F p ) is surjective. So we consider the surjective map

Π p : B(n) ρn -→ Sp(2l, Z) -→ Sp(2l, F p ) π -→ P Sp(2l, F p ) (10)
We consider the image probability Π p µ of µ via Π p on P Sp(2l, F p ). We recall facts on random walks on finite groups.

Theorem 5. (see for exemple [Di]) Let G be a finite group and µ a probability measure on G such that 1. G is generated as a semigroup by supp(µ)

2. supp(µ) is not included in a coset of G by a normal subgroup H of G.
Then µ ⋆m converges to the equidistributed measure on G as m tends to infinity.

We know ( [As]) that, if p > 5, the group P Sp(2n, F p ) is simple so we can apply Theorem 5 and derive Lemma 1. If m tends to infinity, (Π p µ) ⋆m converges to the equidistributed measure on P Sp(2n, F p ).

We now investigate the density of the zero set in Sp(2l, F p ) of the polynomial P appearng in Theorem 3. We write P in terms of all the multi-indices, P (M) = a I m I , where the a I 's belong to Z. If p is an integer, we denote by P p its reduction modulo p.

Since P is not identically zero, we derive an infinite set P of prime numbers p such that P p is not identically zero. To prove the theorem, we need to estimate

|P -1 p (0)| |Sp(2l, F p )| (11)
We recall (see for example [O'M] or [wi]) that

|Sp(2n, Z p )| = n m=1 (p 2m -1)p 2m-1 ≥ 1 2 2n 2l s=1 p s = p 2l 2 +l 2 2n (12)
To estimate |P -1 p (0)|, we use the following result by Lachaud and Rolland Theorem 6. ([L-R]) Let K be the algebraic closure of F p . Let X be an algebraic variety of K n of dimension m which is the zero set of a family of polynomials (f 1 , ..., f r ).

Let d i = deg(f i ). Then |X ∩ F n p | ≤ d 1 ...d r q m
We view P p as a polynomial with coefficients in K and let X = Sp(2l, K)∩ P -1 p (0). We know that for a field K, Sp(2l, K) is irreducible as an algebraic variety ( [Mi]), hence X is of dimension strictly smaller that Sp(2l, K), i.e. dim(X) ≤ 2l 2 + l -1. Thus, for some constant C 1 (2l, P ) depending on 2l and on the degree of P ,

|X ∩ F 4l 2 p | ≤ C 1 (2l, P )p 2l 2 +l-1 (13)
Putting ( 12) and ( 13) together, we derive that

|P -1 p (0)| |Sp(2l, F p )| ≤ C 2 (2l, P ) p (14)
for some constant C 2 (2l, P ). We conclude the proof of Theorem 3. Given a ǫ > 0, we pick a prime number

p in P such that 2 C 2 (2l, P ) p < ǫ (15)
There exists an integer k 0 such that for every k > k 0 , we have

µ ⋆k (P -1 (0)) ≤ (Π p µ) ⋆k (P -1 p (0)) < ǫ.
4 The signature of 3-braids: proof of Corollary 2

4.1 Preliminaries: the Gambaudo-Ghys formula for the signature of a 3-braid ([G-G])

4.1.1 The reduced Burau representation for t = -1

We let B 3 be the 3-braid group and we denote by

B -1 : B(3) -→ SL(2, Z)
the reduced Burau representation of the braid group B(3) for t = -1. Because there are different expressions in the literature, we recall the ones [G-G] uses:

s 1 = B -1 (σ 1 ) = 1 0 -1 1 s 2 = B -1 (σ 2 ) = 1 1 0 1 (16)
We recall (see for example [K-T]):

Fact 1. The center of B(3) is the cyclic group generated by the twist

∆ = (σ 1 σ 2 σ 1 ) 2 . Fact 2. • B -1 is surjective • B -1 (∆) = -Id
• The kernel of B -1 is the cyclic group generated by ∆ 2 .

The Meyer cocycle and the formula for the signature

If β is a braid, we denote its closure by β. For two 3-braids, α and β, [G-G] proves

sign( α.β) = sign( α) + sign( β) -Meyer(B -1 (α), B -1 (β)) ( 17
)
where Meyer ∈ H 2 (SL(2, R)) is the Meyer cocycle. We recall its description by

[G-G]. If γ 1 , γ 2 ∈ SL(2, Z), we let E γ 1 ,γ 2 = Im(γ -1 1 -Id) ∩ Im(γ 2 -Id) ( 18
)
For a vector e in E γ 1 ,γ 2 , we take

v 1 , v 2 in R 2 such that e = γ -1 1 (v 1 ) -v 1 = v 2 -γ 2 (v 2 )
and define the quadratic form

q γ 1 ,γ 2 = Ω(e, v 1 + v 2 ) ( 19
)
where Ω is the standard symplectic form on R 2 . Then Meyer(γ 1 , γ 2 ) is the signature of q γ 1 ,γ 2 .

Proof of Corollary 2

Since σ i and βσ j β -1 close in trivial knots, sign(σ i ) = sign(βσ j β -1 ) = 0; hence sign(σ i βσ j β -1 ) = Meyer(σ i , βσ j β -1 ) (20)

Lemma 2. Let i, j ∈ {1, 2} and β ∈ B(3). If Meyer(σ i , βσ j β -1 ) = 0, then one of the coefficients of the matrix B -1 (β) is zero.

Proof. Recall that Meyer(σ i , βσ j β -1 ) is the signature of a quadratic form on the space E σ i ,βσ j β -1 , defined in §4.1.2; thus, if Meyer(σ i , βσ j β -1 ) is non zero, the space E σ i ,βσ j β -1 contains a non-zero element, i.e. there exist non zero v 1 , v 2 such that

(s -1 i -Id)(v i ) = β • (Id -s j )[β -1 (v 2 )] (21)
If we denote by e 1 , e 2 the canonical basis of R 2 , we notice that

Im(s ±1 1 -Id) = Re 2 Im(s ±1 2 -Id) = Re 1 (22)
It follows that the (LHS) of ( 21) is parallel to e 1 or e 2 and the (RHS) is parallel to β(e 1 ) or β(e 2 ). Thus, for some h, k ∈ {1, 2}, β(e h ) is parallel to e k which implies that one of the coefficients of the matrix of β is zero.

Lemma 3. Let M = a b c d ∈ SL(2, Z). The matrix Ms 1 M -1 s 1 (resp. Ms 2 M -1 s 2 , Ms 1 M -1 s 2 , Ms 2 M -1 s 1 ) is hyperbolic unless b 2 ≤ 4 (resp. c 2 ≤ 4, d 2 ≤ 4, a 2 ≤ 4) (23)
Proof. Compute the traces, for example T race(Ms 2 M -1 s 1 ) = 2 -a 2 .

We can now conclude the proof of Corollary 2. We let

β be a 3-braid such that ∀i, j ∈ {1, 2} | B -1 (β) ij | > 2 (24)
Then sign(σ a βσ b β -1 ) = 0 (Lemma 2) and

σ a βσ b β -1 is hyperbolic (Lemma 3). Thus Meyer(B -1 (σ a βσ b β -1 ), B -1 ((σ a βσ b β -1 ) n ) = 0 (Fact 3 of §4.1.2). It follows that, for every n, sign((σ a βσ b β -1 ) n ) = 0.
On the other hand, Theorem 3 tells us that the braids not verifying (24) form a transient set for the right random µ-walks on B(3).

5 The Lissajous toric knots 5.1 Description 5.1.1 The knots K(N, q, p)

In [S-V], the authors exhibited Lissajous toric knots as one of the classes of boundaries of minimal disks in the 4-ball with a branch point at the origin. Before that, Lamm investigated these knots in connection with billiards in the solid torus ( [L-O]). If N, q, p are integers, with (N, q) = (N, p) = 1, the K(N, q, p) Lissajous toric knot is defined in a 3D-cylinder as

F N,q,p : [0, 2π] -→ S 1 × R 2 F N,q,p : θ → (e N iθ , sin(qθ), cos(pθ + α)) (25)
for a phase α. Endow R 3 with a coordinate system (x, y, z) and take the Lissajous curve C p,q : θ → 0, 2 + sin(qθ), cos(pθ + α) ; then the knot K(N, q, p) is described in R 3 by a point travelling along C p,q while C p,q rotates N times along the vertical axis Oz. We recall a few facts Fact 4. ([S-V]) For a finite number of phases α's the expression in (25) gives us singular crossing points. Otherwise (25) defines a knot and up to mirror transformation, its knot type does not depend on the phase α.

Thus we drop the phase α in (25) and we just talk of a knot K(N, q, p) defined up to mirror symmetry.

5.1.2

The braids B(N, q, p)

The knot K(N, q, p) has a natural N-braid representation B(N, q, p). Fact 5. ([S-V]) Let p, q, d be three positive integers all coprime with N and such that (p, q) = 1. Moreover assume (without loss of generality) that q is odd. Then B(N, dq, dp) = B(N, q, p) d and there exists a braid Q N,q,p such that

B(N, q, p) = Q N,q,p σ ǫ(2) 2 Q -1 N,q,p σ ǫ(1) 1 (26)
with ǫ(1), ǫ(2) ∈ {-1, 1}. If 2N divides p + q or p -q, ǫ(1) = ǫ(2) = 1 so the braid is quasipositive: for N = 3, this happens if and only if q and p are both odd.

The braid ( 26) is a symmetric union as defined by [La] thus: Fact 6. If N, q, p are all mutually prime, K(N, q, p) is a ribbon knot.

See also [L-O] and [S-V].

The signature of the Lissajous toric knots

For N = 3, a Lissajous toric knot has zero signature unless it is a torus knot. More precisely, Theorem 7. Let q, p, n be three positive integers, none of them divisible by 3 and q, p mutually prime. Let K(3, nq, np) be the corresponding Lissajous toric knot.

1. If q and p are both odd, up to mirror image, K(3, nq, np) is a quasipositive knot with g

4 (K(3, nq, np)) = d -1 (27)
and verifying one of the following (a) the signature of K(3, nq, np) is zero (b) B(3, q, p) = σ 2 σ 1 so K(3, q, p) is a trivial knot and K(3, nq, np) is a (3, n)-torus knot.

2. If q and p have different parities, K(3, nq, np) is isotopic to its mirror image, so it has zero signature.

REMARK 1. The Lissajous toric knots K(3, n, n) are just the (3, n) torus knots so clearly they are in the case 1 (b) of Theorem 7, but they are not the only ones. For example if p = q + 6 the knot K(3, q, q + 6) (e.g. K(3, 7, 13)) is represented by the braid B(3, q, q + 6) = σ 2 σ 1 so it is trivial and, for a positive integer n, B(3, nq, n(q + 6)) = (σ 2 σ 1 ) n so K(3, nq, n(q + 6)) is a (3, n)-torus knot. In §6.2, we see that at least for q's up to 100, the majority of knots of 1. in Theorem 7 1. are in the case 1. (a) of that Theorem.

An example is K(3, 5, 7) which is, up to mirror image, the knot 10 155 in the Rolfsen classification and its braid is

B(3, 5, 7) = σ 2 σ -1 1 σ 2 σ -1 1 σ 2 σ 1 σ -1 2 σ 1 σ -1 2 σ 1 (28)

Proof of Theorem 7

The proof depends on the parities of q and p; since these numbers are mutually prime, either they have the same parity or they are both odd.

5.

3.1 1st case: q and p are both odd Up to mirror symmetry, the braid B(3, q, p) is of the form (Fact 5) B(3, q, p) = Q 3,q,p σ 2 Q -1 3,q,p σ 1 with ([S-V]) Q 3,q,p = σ ...σ λ(q-2) 2 σ λ(q-1) 1

(29)

• P = (±1) 0 1 -1 h = (±1)s 1-h 1 s 2 s 1 Thus Q = s -h 1 s -h 2 and similarly to above, β = σ -h 1 (σ 2 σ 1 )σ h 1 . Again we are in Case 1. (b) of Theorem 7.

• P = (±1) 1 1 h h + 1 = (±1)s -h 1 s 2 Then Q = s -h 1 s 2 s -1 1 s h 2 and β = σ -h 1 (σ 2 σ -1 1 σ 2 σ 1 σ -1 2 σ 1 )σ -h 1
This braid β closes into a link with 3 components, thus it cannot be the braid of a knot.

• P = (±1) 1 -1 h 1 -h = s -h 1 s -1 2 and β = σ -h 1 (σ -1 2 σ 1 σ 2 )(σ -1 1 σ 2 σ 1 )σ h 1
Again β closes in a link with 3 components.

2nd case: q and p have different parities

The knot is preserved by an isometry which is a rotation in one plane and a symmetry in its orthogonal plane, thus the knot is isotopic to its mirror image and has zero signature.

To spell this out, we assume that q is odd and p is even. The knot K(N, dq, dp) is defined by the function F N,dq,dp (see (25) above), so we can also define it by the function

Fact 3 .

 3 If γ ∈ SL(2, Z) is hyperbolic (i.e. |tr(γ)| > 2), then for two positive integers a, b, Meyer(γ a , γ b ) = 0 REMARK. Since a generic element of SL(2, Z) is hyperbolic, it follows from Fact 3 that for almost every braid β ∈ B(3), sign(β n ) = nsign(β).

θ

  → F N,dq,dp (θ + π d ) = (e iN θ e iN π d , -sin(dqθ), cos(dpθ + α))thus K(N, dq, dp) is invariant under the transformation  reverses the orientation on R 4 and on S 3 .

Acknowledgements

We thank Tahl Nowik and Chaim Even-Zohar for reading an earlier draft of this paper, Misha Brandenbursky for very helpful conversation on braids and quasimorphisms and Florence Lecomte for necessary advice on algebraic geometry over finite fields.

where λ is an expression with values in {-1, 1} verifying λ(k) = -λ(q -k) so we rewrite Q 3,q,p and introduce the braid P :

We have defined braids and we now turn to their images under the Burau representation B -1 . We let

We notice that the s i 's of ( 16) verify

and derive

If P verifies ( 33), it has one of the following forms, for some integer h.

1 so β is conjugate to σ 2 σ 1 and the knot is as in 1. (b) of Theorem 7.

Statistics

Random quasipositive braids

The following estimates were done with SAGE.

We define

It follows from the previous discussion that, if a 3-braid β belongs to Z, then for all positive integers n, we have

Let W be the µ-random walk on B(3) given by the probability µ on B(3)

Here are the probabilities p(n) that W hits Z at the n-th step, for n ≤ 12. n 1 2 3 4 5 6 7 8 9 10 11 12 p(n) 0 0 0.06 0.11 0.17 0.22 0.27 0.32 0.36 0.41 0.45 0.48 .

Lissajous toric knots with 3 strands

We investigate here the proportion of knots which verify (a) or (b) of 1. of Theorem 7. 1. First we prove Lemma 5. To get a full list of the K(3, q, p)'s for a given q, it is enough to take p ∈ [q + 1, 2q]

Proof. If q and p are odd, mutually prime and not divisible by 3, we use the representation of K(3, q, p) from (5.3.1). The expression of λ is based on the Bezout identity:

for two integers A and B. Then, up to mirror transformation

It follows from ( 36) that, if we replace p by p + q, the expression of λ remains the same.

For a given q, we let P (q) = {p ∈ [q + 1, 2q], such that (2, p) = (3, p) = 1} S(q) = {p ∈ P (q), such that K(3, q, p) verifies 1) (a) of Theorem 7}

The table below gives us, for a given q, the percentage of elements of P (q) which belong to S(q): i.e. the percentage of K(3, q, p)'s for which every K(3, nq, np), n ∈ N, has zero signature. The percentages have been rounded to the integer part.


Paul Rouquette

Claude Amra

Myriam Zerrad

Hélène Krol

Catherine Grezes-Besset

Karine Mathieu

Catherine Grezes

Laser-induced thermal effects in optical interference coatings

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Laser-induced thermal effects in optical interference coatings

Optical interference coatings are commonly used for a large panel of optical components for Space applications. Once submitted to a laser illumination, a small amount of the laser energy is absorbed by the filter materials, which induces a rising temperature inside the component and therefore a change in its thermal radiation. These laser-induced thermal effects are a major issue in the field of Space communication or defense. In this paper, we develop a theoretical model to obtain the temporal and spatial evolution of these effects for a wide range of optical sources and components. In order to obtain the laser induced temperature, the heat equation with the absorption density of the coatings for the source term is solved with an original analytical method based on the Fourier Transform and an analogy between optical propagation and heat diffusion. This allows us not only to obtain the spatial and temporal evolution of the temperature as a function of laser and components parameters, but also to determine laws to predict the maximum temperature elevation, which is a major information for the user and enables to study the nature, thermal or dielectric, of the laser-induced damage threshold. The rising temperature inside the filter will modify its thermal radiation. This thermal effect is modelized by representing the thermal agitation of charged particles by electric current densities. These currents, which are related to the temperature elevation by statistical tools, are then introduced in Maxwell's equations to get the spectral, temporal, and angular patterns of the thermal radiation of arbitrary optical filters submitted to different optical sources. The patterns of thermal radiation of optical interference coatings are generally closed to those of the blackbody apart from their very low level due to the low absorption of the materials used. Thanks to our modelling and its similarities with the models used in optical design, we present planar multilayers that enable to shape their thermal radiation. These structures, based on the concept of zero admittance layer (ZAL), concentrate their thermal radiation in a very small spectral or angular bandwidth. They are an interesting alternative to 2D structures such as gratings or meta-surfaces, or to photonic crystals. This work was carried out within the framework of the joint laboratory LABTOP between the Fresnel Institute and CILAS ArianeGroup with the support of the ANRT and CNES.

P. Rouquette 1,2 , C. Amra 1 , M. Zerrad 1 , H. Krol 2 , C. Grèzes-Besset 2 , Karine Mathieu 3 1

  Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Faculté des Sciences -Campus Saint Jérôme, Avenue Escadrille Normandie-Niemen, 13397 Marseille, France 2 CILAS ArianeGroup, 600 avenue de La Roche Fourcade, Pôle Alpha-Sud, 13400 Aubagne

	3 CNES, 18 avenue Edouard Belin 31401 Toulouse, France


Séverine Olivier
email: sev-olivier@orange.fr

Hanna Diounou
email: hanna.diounou@inserm.fr

Marc Foretz
email: marc.foretz@inserm.fr

Sandra Guilmeau
email: sandra.guilmeau@inserm.fr

Noëmie Daniel
email: noemie.daniel@inserm.fr

André Marette
email: andre.marette@criucpq.ulaval.ca

Malvyne Rolli-Derkinderen

Benoit Viollet
email: benoit.viollet@inserm.fr

Malvyne Derkinderen
email: malvyne.derkinderen@univnantes.fr

L'activité de l'AMPK garante de la barrière épithéliale de l'intestin

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La barrière épithéliale intestinale (BEI) présente une double fonction au sein du système gastro-intestinal assurée par l'intégrité de la muqueuse intestinale , elle permet l'absorption des nutriments protège l'organisme contre les atteintes de l'environnement extérieur (pathogènes et toxines présents dans le compartiment luminal) assurée par l'intégrité de la muqueuse intestinale. Les cellules épithéliales intestinales organisées en monocouche sont étroitement liées entre elles grâce au réseau des protéines intercellulaires, notamment celles des jonctions serrées qui régulent principalement la perméabilité paracellulaire. Des dysfonctionnements de la fonction barrière de l'intestin, et notamment l'altération des jonctions serrées, sont caractérisés par une augmentation de la perméabilité intestinale ou "fuite intestinale", et ont été associés à la sévérité de nombreuses maladies intestinales, comme les maladies inflammatoires chroniques de l'intestin (MICI) comprenant la maladie de Crohn ou la rectocolite hémorragique [START_REF] Mankertz | Altered permeability in inflammatory bowel disease: pathophysiology and clinical implications[END_REF]. De manière intéressante, une augmentation de la perméabilité intestinale est également associée à des pathologies en dehors de la sphère digestive, comme les maladies neurodégénératives, et en particulier avec de nombreux facteurs de risque de maladies métaboliques. Les altérations de la BEI contribuent par exemple au développement de l'obésité et de l'inflammation systémique de bas grade qui l'accompagne [START_REF] Genser | Increased jejunal permeability in human obesity is revealed by a lipid challenge and is linked to inflammation and type 2 diabetes[END_REF]. Le renforcement de la BEI constitute donc une approche thérapeutique intéressante pour limiter les conséquences pathologiques d'une hyper-perméabilité intestinale.

Cependant, cette stratégie nécessite une meilleure compréhension des mécanismes de régulation de la BEI et l'identification de nouvelles cibles thérapeutiques.

L'AMPK (AMP-activated protein kinase) est une protéine kinase principalement connue pour son rôle de senseur cellulaire dans la régulation de l'homéostasie énergétique. Suite à un déficit de la balance énergétique cellulaire, l'AMPK est activée et initie des changements métaboliques contribuant à une reprogrammation du métabolisme qui est caractérisée par l'inhibition des processus anaboliques non essentiels consommateurs d'ATP et la stimulation des voies cataboliques produisant de l'ATP permettant ainsi le rétablissement de la balance énergétique cellulaire [START_REF] Hardie | AMP-activated protein kinase: maintaining energy homeostasis at the cellular and whole-body levels[END_REF]. Cependant, son action ne se limite pas au seul contrôle du métabolisme énergétique. En effet, plusieurs études ont mis en lumière son rôle, indépendamment de sa fonction de senseur métabolique, dans la régulation de l'assemblage des jonctions serrées et de la perméabilité épithéliale, comme initialement démontré à l'aide de cellules épithéliales rénales [4; 5]. Des résultats similaires ont été rapportés pour les cellules épithéliales intestinales humaines Caco-2, issues d'un adénocarcinome de côlon, montrant que l'activation pharmacologique de l'AMPK améliore l'assemblage des jonctions serrés suite à un dommage, tandis que son inhibition provoque une altération de la localisation de la protéine ZO-1 aux jonctions serrées et un défaut de résistance électrique trans-épithéliale (TEER) et de la fonction barrière [START_REF] Olivier | AMPK Activation Promotes Tight Junction Assembly in Intestinal Epithelial Caco-2 Cells[END_REF][START_REF] Sun | AMPK improves gut epithelial differentiation and barrier function via regulating Cdx2 expression[END_REF][START_REF] Ghosh | The stress polarity signaling (SPS) pathway serves as a marker and a target in the leaky gut barrier: implications in aging and cancer[END_REF]. Dans une nouvelle étude, nous avons cherché à approfondir notre compréhension du rôle régulateur de l'AMPK sur l'homéostasie épithéliale intestinale in vivo [START_REF] Olivier | Deletion of intestinal epithelial AMP-activated protein kinase alters distal colon permeability but not glucose homeostasis[END_REF]. En utilisant des modèles murins déficients pour les sous-unités catalytiques AMPKα1 et AMPKα2 de l'AMPK spécifiquement dans les cellules épithéliales intestinales, nous avons étudié les conséquences d'une absence d'activité AMPK intestinale sur la perméabilité, la susceptibilité à l'inflammation et la composition du microbiote intestinal (Figure 1). L'analyse de l'intégrité de la barrière intestinale chez les souris dépourvues d'AMPK intestinale a été réalisée en mesurant la perméabilité transcellulaire et paracellulaire des différents segments de l'intestin grêle et du côlon en chambre de Ussing. De manière intéressante, nous avons observé une perméabilité plus élevée exclusivement dans le côlon distal, indiquant une perturbation de la barrière épithéliale dans ce segment intestinal.

Toutefois, ce phénotype n'est pas lié à une différence dans la réponse inflammatoire comme le montrent les niveaux équivalents de cytokines pro-inflammatoires dans le côlon ou de lipocaline 2 dans les féces des animaux dépourvus ou non d'AMPK intestinale. Bien que l'activation de l'AMPK ait été associée à une diminution de la perméabilité paracellulaire in vitro conjointement à l'augmentation des quantités des protéines de jonctions serrées ZO-1, claudine et occludine, nous n'avons observé aucune modification de leur expression ou distribution en l'absence de l'AMPK intestinale in vivo. Par contre, ce défaut de perméabilité du côlon distal est associé à la réduction de la phosphorylation de la protéine d'échafaudage girdine qui participe à la protection de l'intégrité de l'épithélium intestinale en stabilisant les jonctions serrées [8; 9]. Ces résultats sont cohérents avec le fait que la girdine constitue un substrat de l'AMPK, comme le démontre la perte de sa phosphorylation sur le résidu Ser245 en réponse à l'activation pharmacologique de l'AMPK dans un modèle de cellules polarisées dérivées d'entéroïdes provenant du côlon de souris dépourvues en AMPK intestinale. La monocouche de cellules dépourvues d'AMPK présentent une altération de la fonction barrière avec une TEER diminuée en parallèle du défaut de phosphorylation de la girdine [START_REF] Ghosh | The stress polarity signaling (SPS) pathway serves as a marker and a target in the leaky gut barrier: implications in aging and cancer[END_REF]. Bien qu'il existe un lien entre l'hyperperméabilité intestinale et le développement de l'obésité et du diabète de type 2 [START_REF] Genser | Increased jejunal permeability in human obesity is revealed by a lipid challenge and is linked to inflammation and type 2 diabetes[END_REF], nous n'avons observé aucune aggravation de l'adiposité, de l'intolérance au glucose, de défaut de sécrétion de GLP-1 ou de la résistance à l'insuline chez les animaux dépourvus d'AMPK intestinale même après un challenge avec un régime riche en graisse. Par ailleurs, nous montrons que l'AMPK intestinale n'est pas nécessaire pour améliorer la tolérance au glucose après administration de l'anti-diabétique oral metformine.

En conclusion, l'AMPK joue un rôle important dans la régulation de la fonction barrière, en particulier au niveau du côlon distal. L'AMPK intestinale participe au contrôle de la perméabilité intestinale en modulant la phosphorylation de la girdine, garant de l'intégrité des jonctions serrées (Figure 1). Alors que nos données soutiennent un rôle de l'AMPK intestinale dans le maintien de l'homéostasie intestinale, nous avons constaté que l'AMPK intestinale n'était pas nécessaire pour la régulation de l'homéostasie du glucose ou l'effet antihyperglycémiant de la metformine. Nos travaux soulignent également une interaction complexe entre le microbiote intestinal et l'AMPK de l'hôte, l'absence d'AMPK intestinale étant associée à une dysbiose (Figure 1). En accord, il a été récemment suggéré que les modifications de la diversité et la composition microbienne induites par un régime riche en fibres seraient perdues lorsque l'activité de l'AMPK est inhibée [START_REF] Li | Potential correlation between carbohydrate-active enzyme family 48 expressed by gut microbiota and the expression of intestinal epithelial AMP-activated protein kinase β[END_REF]. Ainsi, il serait intéressant d'examiner si le ciblage de la voie de signalisation AMPK par des approches pharmacologiques et/ou nutritionnelles (régimes riches en fibres) peut conduire à de nouvelles thérapeutiques pour renforcer la barrière intestinale, en particulier dans le contexte de pathologies associées à une hyper-perméabilité intestinale.

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Les auteurs déclarent n'avoir aucun lien d'intérêt concernant les données publiées dans cet article.

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  'analyse du microbiote fécal chez les animaux dépourvus d'AMPK intestinale a révélé un changement significatif dans la composition de la communauté bactérienne intestinale. L'absence d'AMPK intestinale est corrélée à une abondance plus importante des bactéries de la famille des Clostridiales et Desulfovibrionaceae. En accord avec une altération de la barrière intestinale, les bactéries de la famille des Desulfovibrionaceae sont retrouvées de manière plus importante chez les patients atteints de colite ulcéreuse et ont été associées à une réduction de l'épaisseur des muqueuses, ce qui facilite le contact entre les antigènes bactériens et le système immunitaire de la muqueuse intestinale [10]. D'autres espèces de la flore bactérienne intestinale (par exemple, Clostriales, Dubosiella, Roseburia, Alloprevotella, Enterorhabdus et Anaerovorax) sont également augmentées chez les souris dépourvues d'AMPK intestinale. Cependant, nous n'avons pas observé de changement majeur sur la composition des populations intestinales chez les souris nourries avec un régime riche en graisses. Le rôle de ces bactéries et surtout la fonction du microbiote intestinal en lien avec l'AMPK reste donc encore à démontrer, par exemple via des expériences de transfert de microbiote intestinal.

Figure 1 :

 1 Figure 1: Impact de la délétion de l'AMPK intestinale chez la souris sur l'intégrité de la barrière intestinale, la phosphorylation de la girdine et la composition du microbiote au niveau du colon.