MMARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS FATHI BEN ARIBI
Abstract.
We study the effect of Markov moves on L -Burau maps of braids,in order to construct link invariants from these maps with a process mirroringthe well-known Alexander-Burau formula.We prove such a Markov invariance for L -Burau maps going to the groupsof the braid closures or deeper. In the case of the link group, the correspondinglink invariant was known to be the L -Alexander torsion by a previous resultof A. Conway and the author.Furthermore, we find two counter-examples, meaning two families of L -Burau maps that cannot yield link invariants in this way. The proofs use rela-tions between Fuglede-Kadison determinants, Mahler measures, and randomwalks on Cayley graphs, and the works of Boyd, Bartholdi and Dasbach-Lalin.All of this suggests that twisted L -Alexander torsions are the only linkinvariants we can hope to construct from L -Burau maps with such a process. Introduction
The L -Burau maps and reduced L -Burau maps of braids were introduced in2016 in [BAC] by A. Conway and the author. These maps generalise the Buraurepresentation of braid groups and are indexed by a positive real number t and agroup epimorphism γ of a free group of finite rank. In a sense, all the L -Buraumaps are contained between the Burau representation and the Artin injection ofthe braid group B n in the automorpishm group Aut( F n ) of the free group F n .Moreover, we proved in [BAC] that for a braid β ∈ B n , its image by the reduced L -Burau map B (2) t,γ β indexed by the epimorphism γ β : F n → G β ∼ = π (cid:16) S \ ˆ β (cid:17) yields the L -Alexander torsion of the braid closure ˆ β (a link invariant due toLi-Zhang and Dubois-Friedl-L¨uck [LZ, DFL]), and thus that B (2) t,γ β contains deeptopological information about β such as the hyperbolic volume or the genus of ˆ β .It is then natural to wonder whether L -Burau maps associated to other epimor-phisms can similarly provide link invariants and detect topological information ofthe braid, and this article provides a partial positive answer to this question.We first introduce the notion of Markov-admissibility of a family Q of groupepimorphisms Q β : F n ( β ) (cid:16) G Q β indexed by braids β ∈ t n (cid:62) B n , in Section 3.Roughly speaking, we say that such a family is Markov-admissible when for anytwo braids α, β that have isotopic closures (and thus are related by Markov moves),the associated epimorphisms Q α and Q β descend “to the same depth” and arerelated by a sequence of commutative diagrams. Markov admissibility appears tobe a necessary condition in order to construct Markov functions and link invariantsfrom general families of epimorphisms indexed by braids.In this paper we focus on a specific candidate for being a Markov function,namely F Q := t n (cid:62) B n → F ( R > , R > ) / { t t m , m ∈ Z } β t det rG Qβ (cid:16) B (2) t,Q β ( β ) − Id ⊕ ( n − (cid:17) max(1 , t ) n , a r X i v : . [ m a t h . G T ] J a n FATHI BEN ARIBI where Q is a Markov-admissible family of epimorphisms and det rG Qβ is the regularFuglede-Kadison determinant , a version of the determinant for infinite-dimensional G -equivariant operators on ‘ ( G ) such as the L -Burau maps (see Section 2 for adefinition).The first main result of this article is the following theorem: Theorem 1.1 (Theorem 4.1) . Let Q be a Markov-admissible family of epimor-phisms that descends to the groups of the braid closures or deeper. Then F Q is aMarkov function, and thus defines an invariant of links. As detailed in Section 4, to prove Theorem 1.1 we study how Markov moveson braids modify reduced L -Burau maps and we use properties of the Fuglede-Kadison determinant.Part of Theorem 1.1 was already proven in [BAC], but without consideringMarkov invariance. Indeed, for Q the family that exactly descends to the groups ofthe braid closures G β , [BAC, Theorem 4.9] directly linked F Q to the L -Alexandertorsions of the braid closures, as a variant of the well-known Alexander-Burau for-mula; and since the L -Alexander torsions were already known link invariants,Markov invariance was reciprocally guaranteed, and carefully studying Markovmoves was unnecessary.However, as stated in Theorem 1.1, the methods of the current article pro-vide several new link invariants, which appear (unsurprisingly) to be twisted L -Alexander torsions of links.Theorem 1.1 is not an equivalence in the sense that F Q could theoretically be aMarkov function for other families Q , but the specific convenient cancellations thatoccur in matrix coefficients in the proof of Theorem 1.1 make this seem unlikely. Asfurther evidence, we establish two families Q such that F Q is not a Markov function,in the second main result of this article: Theorem 1.2 (Theorems 5.1 and 5.5) . Let Q be either the family of identities ofthe free groups or the family of abelianizations of the free groups. Then F Q is nota Markov function. Our main tools to prove Theorem 1.2 are relations between Fuglede-Kadisondeterminants (which are technical to define and difficult to compute), Mahler mea-sures of polynomials (notably studied by Boyd [Bo]) and combinatorics on Cayleygraphs (developed in [Ba, DL]). Along the way, we notably compute new values forFuglede-Kadison determinants of operators over the free groups (see Proposition5.3).As the reader will probably agree, the initial question (how to build link in-variants from L -Burau maps) is still far from answered. We restricted ourselvesto studying the most intuitive form of (potential) Markov function F Q , and foundthat twisted L -Alexander torsions appeared to be the best link invariants we couldobtain with it. This last point is unsurprising considering the form of F Q and itsnatural connection with the Alexander polynomial and its variations.However, there may well be new link invariants to discover via other functions ofthe L -Burau maps, and we hope that our computations of how these maps changeunder Markov moves can be of use for future research in this vein.The present article arose as a part of a wider project in collaboration with C.Anghel aiming to construct new knot invariants from L -versions of the Burau andLawrence representations of braid groups. To attain this end, studying the influenceof Markov moves on L -Burau maps appears to be a natural step.The article is organised as follows: in Section 2 we recall preliminaries on braidgroups and L -invariants; in Section 3 we introduce the notion of a Markov-admissible family of group epimorphisms with several examples; in Section 4 we ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 3 state and prove Theorem 4.1 on Markov invariance; finally, in Section 5 we presentcounter-examples to Markov invariance and new computations of Fuglede-Kadisondeterminants. Acknowledgements
The author was supported by the FNRS in his position at UCLouvain. We thankCristina Anghel-Palmer and Anthony Conway for helpful discussions.2.
Preliminaries
We will mostly follow the conventions of [BAC] and [Lu].2.1.
Braid groups.
The braid group B n can be seen as the set of isotopy classes oforientation-preserving homeomorphisms of the punctured disk D n := D \{ p , . . . , p n } which fix the boundary pointwise. Recall that B n admits a presentation with n − σ , σ , . . . , σ n − following the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 for each i ,and σ i σ j = σ j σ i if | i − j | >
2. Topologically, the generator σ i is the braid whose i -thcomponent passes over the ( i + 1)-th component. x x x z Figure 1.
The punctured disk D .Fix a base point z of D n and denote by x i the simple loop based at z turning oncearound p i counterclockwise for i = 1 , , . . . , n (see Figure 1). The group π ( D n ) canthen be identified with the free group F n on the x i . If H β is a homeomorphism of D n representing a braid β , then the induced automorphism h β of the free group F n depends only on β . It follows from the way we compose braids that h αβ = h β ◦ h α ,and the resulting right action of B n on F n (named the Artin action ) can be explicitlydescribed by h σ i ( x j ) = x i x i +1 x − i if j = i,x i if j = i + 1 ,x j otherwise, h σ − i ( x j ) = x i +1 if j = i,x − i +1 x i x i +1 if j = i + 1 ,x j otherwise.In this paper we will also use a second set of generators of F n , namely g := x , g := x x , . . . , g n := x . . . x n . Looking at Figure 1, g i represents the class of the loop that circles the first i punctures. On these generators, B n acts in the following way: h σ i ( g j ) = ( g i +1 g − i g i − if j = i,g j otherwise, h σ − i ( g j ) = ( g i − g − i g i +1 if j = i,g j otherwise,where we use the convention g := 1. FATHI BEN ARIBI
Fox calculus.
Denoting by F n the free group on x , x , . . . , x n , the Fox de-rivative ∂∂x i : Z [ F n ] → Z [ F n ] (first introduced in [Fox]) is the linear extension ofthe map defined on elements of F n by ∂x j ∂x i = δ i,j , ∂x − j ∂x i = − δ i,j x − j , ∂ ( uv ) ∂x i = ∂u∂x i + u ∂v∂x i . The following formula is sometimes referred to the fundamental formula of Foxcalculus:
Proposition 2.1.
Let u ∈ F n and (cid:15) : ZF n → Z the map defined by (cid:15) : x i .Then: u − (cid:15) ( u ) · n X i =1 ∂u∂x i · ( x i − . Fuglede-Kadison determinant.
In this section we will give short definitionsof the von Neumann trace and the Fuglede-Kadison determinant, without going intotechnical details. More details can be found in [Lu] and [BAC].Let G be a finitely generated group. The Hilbert space ‘ ( G ) is the completionof the group algebra C G , and the space of bounded operators on it is denoted B ( ‘ ( G )). We will focus on right-multiplication operators R w ∈ B ( ‘ ( G )), where R · denotes the right regular action of G on ‘ ( G ) extended to the group ring C G (and even to the rings of matrices M p,q ( C G )).For any element w = a · G + a g . . . + a r g r ∈ C G , the von Neumann trace tr G of the associated right multiplication operator is defined astr G ( R w ) = tr G (cid:0) a Id ‘ ( G ) + a R g . . . + a r R g r (cid:1) := a , and the von Neumann trace for a finite square matrix over C G is given as the sumof the traces of the diagonal coefficients.Now the most concise definition of the Fuglede-Kadison determinant det G of aright-multiplication operator A is probablydet G ( A ) := lim ε → + (cid:18) exp ◦ (cid:18)
12 tr G (cid:19) ◦ ln (cid:19) (( A ⊥ ) ∗ ( A ⊥ ) + ε Id) (cid:62) , where A ⊥ is the restriction of A to a supplementary of its kernel, ∗ is the adjunctionand ln the logarithm of an operator in the sense of the holomorphic functionalcalculus. Compare with [Lu, Theorem 3.14] and Proposition 2.2 below.The following properties concern the classical Fuglede-Kadison determinant det G described in [Lu2, Chapter 3], which is not always multiplicative if one deals withnon injective operators. Moreover, this determinant forgets about the influenceof the spectral value 0, which surprisingly makes it take the value 1 for the zerooperator. More recent articles have used the regular Fuglede-Kadison determinant det rG instead, which is defined for square injective operators, is zero for non injectiveoperators, and is always multiplicative. In this paper we will work with both typesof determinants, but the reader should be reassured that most of the statements wewill make remain unchanged while replacing one determinant with the other (upto assumptions on injectivity usually). Similarly, the statements of the followingProposition 2.2 admit immediate variants with det rG . Proposition 2.2 ([Lu2]) . Let G be a countable discrete group and let A, B, C, D ∈ t p,q ∈ N R M p,q ( C G ) be general right multiplication operators. The Fuglede-Kadison determinant satisfiesthe following properties: ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 5 (1) (multiplicativity) If A, B are injective, square and of the same size, then det G ( A ◦ B ) = det G ( A ) det G ( B ) . (2) (block triangular case) If A, B are injective and square, then det G (cid:18) A C B (cid:19) = det G ( A ) det G ( B ) , where C has the appropriate dimensions. (3) (induction) If ι : G , → H is a group monomorphism, then det H ( ι ( A )) = det G ( A ) . (4) (relation with the von Neumann trace) If A is a positive operator, then det G ( A ) = (exp ◦ tr G ◦ ln) ( A ) . (5) (simple case) If g ∈ G is of infinite order, then for all t ∈ C the operator Id − tR g is injective and det G (Id − tR g ) = max(1 , | t | ) . (6) ( × trick) For all A, B, C, D ∈ N ( G ) such that B is invertible, (cid:18) A BC D (cid:19) is injective if and only if DB − A − C is injective, and in this case one has: det G (cid:18) A BC D (cid:19) = det G ( B ) det G ( DB − A − C ) . (7) (relation with Mahler measure) Let G = Z d , and P denote the d -variablepolynomial associated to the operator A ∈ R CZ d . Then det Z d ( A ) = M ( P ) , where M is the d -variable Mahler measure. (8) (limit of positive operators) If A is injective, then det G ( A ) = lim ε → + p det G ( A ∗ A + ε Id) . (9) (dilations) Let λ ∈ C ∗ . Then: det G ( λ Id ⊕ n ) = | λ | n . Remark . If the group G satisfies the strong Atiyah conjecture (see [Lu, Chap-ter 10]), then the right multiplication operator by any non-zero element of C G isinjective, which makes it convenient to apply some parts of Proposition 2.2. Freegroups and free abelian groups are examples of such groups.2.4. L -Burau maps on braids. Let n ∈ N ∗ , t >
0, Φ n : F n (cid:16) Z denote theprojection of all free generators to 1, and γ : F n (cid:16) G denote an epimorphismsuch that Φ n factors through γ . Let κ ( t, Φ n , γ ) : ZF n → R G denote the ringhomomorphism that sends g ∈ F n to t Φ n ( g ) γ ( g ).Then following [BAC], the associated L -Burau map on B n is B (2) t,γ : B n β R κ ( t, Φ n ,γ )( J ) ∈ B (cid:0) ‘ ( G ) ⊕ n (cid:1) , where J = (cid:18) ∂h β ( x j ) ∂x i (cid:19) (cid:54) i,j (cid:54) n is the Fox jacobian of h β for the base of the x i , andthe associated reduced L -Burau map on B n is B (2) t,γ : B n β R κ ( t, Φ n ,γ )( J ) ∈ B (cid:16) ‘ ( G ) ⊕ ( n − (cid:17) , where J = (cid:18) ∂h β ( g j ) ∂g i (cid:19) (cid:54) i,j (cid:54) n − is the Fox jacobian of h β for the base of the g i . FATHI BEN ARIBI
In the remainder of this article we will focus on reduced L -Burau maps.Note that L -Burau maps (reduced or not) admit other definitions as mapsover a certain homology of a cover of the punctured disk (see [BAC] for details).Although these homological definitions may be more natural and more useful forfurther generalizations, the remainder of this article will only use the previousdefinitions via Fox jacobians.The following proposition is an (anti-) multiplication formula for the reduced L -Burau maps. Proposition 2.4 ([BAC]) . For any n, t, γ as above and any two braids α, β ∈ B n ,we have: B (2) t,γ ( αβ ) = B (2) t,γ ( β ) ◦ B (2) t,γ ◦ h β ( α ) . Note that the unreduced L -Burau maps satisfy an identical formula.It follows from Proposition 2.4 that a reduced L -Burau map can be computedfor any braid via knowing the values on the generators σ i of the braid group. Forthe reader’s convenience and since they will be used in the remainder of this article,we now provide the image of the generators σ i : B (2) t,γ ( σ ) = (cid:18) − tR γ ( g g − ) (cid:19) ⊕ Id ⊕ ( n − , B (2) t,γ ( σ i ) = Id ⊕ ( i − ⊕ Id tR γ ( g i +1 g − i ) − tR γ ( g i +1 g − i )
00 Id Id ⊕ Id ⊕ ( n − i − f or < i < n − , B (2) t,γ ( σ n − ) = Id ⊕ ( n − ⊕ Id tR γ ( g n g − n − ) − tR γ ( g n g − n − ) ! , and the images of the inverses σ − i : B (2) t,γ ( σ − ) = − t R γ ( g − ) t R γ ( g − ) Id ! ⊕ Id ⊕ ( n − , B (2) t,γ ( σ − i ) = Id ⊕ ( i − ⊕ Id Id 00 − t R γ ( g i − g − i ) t R γ ( g i − g − i ) Id ⊕ Id ⊕ ( n − i − f or < i < n − , B (2) t,γ ( σ − n − ) = Id ⊕ ( n − ⊕ Id Id0 − t R γ ( g n − g − n − ) ! . Remark . It follows from what precedes and from Proposition 2.2 that for all t >
0, we have det G (cid:16) B (2) t,γ ( σ ± i ) (cid:17) = t ± .3. Markov admissibility of a family of epimorphisms
For each n (cid:62) β ∈ B n , let us denote • n ( β ) := n the number of strands, • h β the (Artin) group automorphism on F n ( β ) (note that β h β is anti-multiplicative), • γ β : F n ( β ) (cid:16) G β the quotient by all relations of the form · = h β ( · ), • ˆ β the closure of β , a link in S , • G ˆ β = π (cid:16) S \ ˆ β (cid:17) the group of the link ˆ β , • Φ n : F n (cid:16) Z the projection of all free generators to 1, ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 7 • ι n : F n , → F n +1 the group inclusion sending the n generators of F n to thefirst n generators of F n +1 . Definition 3.1.
A family Q of group epimorphisms of the form Q = (cid:8) Q β : F n ( β ) (cid:16) G Q β | β ∈ t n (cid:62) B n (cid:9) will be called Markov-admissible if it satisfies the following conditions:(1) For all n (cid:62) α, β ∈ B n , there exists a group isomorphism χ Q β,α : G Q β ∼ → G Q α − βα such that Q α − βα ◦ h α = χ Q β,α ◦ Q β .(2) For all n (cid:62) β ∈ B n and ε ∈ {± } , there exists a group monomorphism σ Q β,ε : G Q β , → G Q σεnβ such that Q σ εn β ◦ ι n = σ Q β,ε ◦ Q β .These two conditions are illustrated in Figure 2. G Q β χ Q β,α ∼ G Q α − βα F n Q β h α ∼ F n Q α − βα G Q β , σ Q β,ε G Q σεnβ F n Q β , ι n F n +1 Q σ εn β Figure 2.
Conditions for Q to be Markov-admissibleRoughly speaking, the epimorphisms in a Markov-admissible family will be com-patible in a way that lets us hope to compute ( L -)knot invariants by using them.Note that less restrictive definitions may be preferred in the future if we find betterways of computing L -objects such as Fuglede-Kadison determinants.We will now present several examples of Markov-admissible families, which areall displayed in Figure 3 for clarity. Moreover, Figure 3 summarizes the results ofSections 4 and 5 concerning Markov invariance ( (cid:88) standing for yes and X for no). F n X Φ n Z (cid:88) γ β G β (cid:88) Ψ β ∼ G ˆ β (cid:88) ψ β Γ ψ β (cid:88) ϕ n Z n X Figure 3.
When does B (2) t,γ yield a map on braids which is invari-ant under Markov moves, for γ : F n (cid:16) G ? Example . The family of identity morphisms Q = (cid:8) id F n ( β ) (cid:9) is Markov-admissible,with χ Q β,α = id F n ( β ) and σ Q β,ε = ι n ( β ) . Example . At the other end of the spectrum, the family Q = (cid:8) Φ n ( β ) (cid:9) is Markov-admissible, with χ Q β,α = σ Q β,ε = id Z . FATHI BEN ARIBI
Example . The family of abelianizations Q = (cid:8) ϕ n ( β ) : F n ( β ) (cid:16) Z n ( β ) (cid:9) is Markov-admissible, with σ Q β,ε the inclusion Z n ( β ) , → Z n ( β )+1 induced by ι n ( β ) , and χ Q β,α thepermutation on the canonical generators of Z n corresponding to the permutationof n ( α ) strands induced by α ∈ B n .The following proposition is an elementary result in group theory, but is statedfor the reader’s convenience. Proposition 3.5.
Let f : G → H be a group homomorphism, and N a normalsubgroup of G . If f is surjective and Ker ( f ) ⊂ N (in particular if f is an isomor-phism), then f induces an isomorphism between G/N and
H/f ( N ) . Let us now consider epimorphisms that descend to the fundamental groups ofthe braid closure complements.
Proposition 3.6.
The family Q = { γ β : F n ( β ) (cid:16) G β } is Markov-admissible. More-over the monomorphisms σ Q β,ε are isomorphisms.Proof. First step: Markov 1:Take n (cid:62) α, β ∈ B n . Note that h α ( Ker ( γ β )) = h α (cid:0) hh h β ( x ) x − ; x ∈ F n ii (cid:1) = hh h βα ( x ) h α ( x ) − ; x ∈ F n ii = hh h α − βα ( y ) y − ; y ∈ F n ii = Ker ( γ α − βα ) , thus it follows from Proposition 3.5 that the isomorphism h α : F n ∼ → F n induces therequired isomorphism χ Q β,α : G β ∼ → G α − βα .Second step: Markov 2:Take n (cid:62) β ∈ B n . Let β + = σ − n ι ( β ) ∈ B n +1 . First notice that h β + ( x j ) x − j = h ι ( β ) ( h σ − n ( x j )) x − j = h ι ( β ) ( x j ) x − j for 1 (cid:54) j (cid:54) n − ,h β + ( x n ) x − n = h ι ( β ) ( h σ − n ( x n )) x − n = h ι ( β ) ( x n +1 ) x − n = x n +1 x − n ,h β + ( x n +1 ) x − n +1 = h ι ( β ) ( h σ − n ( x n +1 )) x − n +1 = h ι ( β ) ( x − n +1 x n x n +1 ) x − n +1 = x − n +1 h ι ( β ) ( x n ) . Hence
Ker ( γ β + ) = hh x n +1 x − n ; ι n ( Ker ( γ β )) ii .We can now define σ Q β, − := (cid:16) G β [ x ] G β [ ι n ( x )] G β + ∈ G β + (cid:17) , where x ∈ F n and [ · ] G is the quotient class in G . Since ι n ( Ker ( γ β )) ⊂ Ker ( γ β + ), σ Q β, − is awell-defined group homomorphism.Let us prove that σ Q β, − is surjective. Let [ y ] G β + ∈ G β + , with y ∈ F n +1 . Let y ∈ ι n ( F n ) be the word constructed from y by replacing all letters x n +1 with x n .Hence [ y ] G β + = [ y ] G β + ∈ Im ( σ Q β, − ) and σ Q β, − is surjective.Let us prove that σ Q β, − is injective. Let [ ι n ( x )] G β + ∈ G β + (with x ∈ F n ) betrivial. Then ι n ( x ) ∈ Ker ( γ β + ) = hh x n +1 x − n ; ι n ( Ker ( γ β )) ii . Thus ι n ( x ) is aproduct of conjugates (in F n +1 ) of terms ( x n +1 x − n ) ± and/or terms in ι n ( Ker ( γ β )).But since ι n ( x ) is a free word without the letter x n +1 , we conclude that the conju-gates of terms ( x n +1 x − n ) ± in it cancel each other. Thus ι n ( x ) ∈ ι n ( Ker ( γ β )) and[ x ] G β = 1. Hence σ Q β, − is injective.Third step: Markov 2 again:Take n (cid:62) β ∈ B n , and let β + = σ n ι ( β ) ∈ B n +1 . The proof is similar as in theSecond step, with the following differences: h β + ( x n ) x − n = h ι ( β ) ( x n ) x n +1 (cid:0) h ι ( β ) ( x n ) (cid:1) − x − n ,h β + ( x n +1 ) x − n +1 = h ι ( β ) ( x n ) x − n +1 ,Ker ( γ β + ) = hh h ι ( β ) ( x n ) x − n +1 ; ι n ( Ker ( γ β )) ii , ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 9 and we replace letters x n +1 with h ι ( β ) ( x n ) in the proof of the surjectivity of σ Q β, . (cid:3) Finally, let us fix notations for epimorphisms that go deeper than the groups ofthe braid closures.
Example . Let (cid:8) ψ β : G β (cid:16) Γ ψ β (cid:9) β ∈t n (cid:62) B n be a family of epimorphisms such that • Φ n ( β ) factors through ψ β ◦ γ β , • χ Q β,α ( Ker ( ψ β )) = Ker ( ψ α − βα ), • σ Q β,ε ( Ker ( ψ β )) = Ker ( ψ σ εn ι ( β ) ).Then it follows from Propositions 3.5 and 3.6 that the family Q = (cid:8) ψ β ◦ γ β : F n ( β ) (cid:16) Γ ψ β (cid:9) is Markov-admissible.4. A sufficient condition for Markov invariance
For Q = { Q β } a Markov-admissible family, and t >
0, we define the function F Q := t n (cid:62) B n → F ( R > , R > ) / { t t m , m ∈ Z } β t det rG Qβ (cid:16) B (2) t,Q β ( β ) − Id ⊕ ( n − (cid:17) max(1 , t ) n , taking values in equivalence classes [ t f ( t )] of selfmaps of R > , up to multiplica-tion by monomials with integer exponents.The following theorem gives a sufficient condition on Q for F Q to be a Markovfunction and thus yield a invariant of links. Theorem 4.1.
Let Q = { ψ β ◦ γ β : F n ( β ) (cid:16) Γ ψ β } be as in Example 3.7. Then F Q isa Markov function, and thus defines an invariant of links, that we will denote T Q . Remark that for Q = { Ψ β ◦ γ β : F n ( β ) (cid:16) G ˆ β } β , where Ψ β denotes an isomorphismbetween G β and G ˆ β , it follows from [BAC] that the link invariant T Q is the L -Alexander torsion.Moreover, it seems that the link invariant T Q (for a general Q of the form ofTheorem 4.1) is a twisted L -Alexander torsion, where the twist is by the generalepimorphism ψ β . To prove this, one would need to adapt the proof of the mainresult of [BAC].To prove Theorem 4.1, we will use two lemmas, one for each Markov move. Lemma 4.2.
Let Q = { ψ β ◦ γ β : F n ( β ) (cid:16) Γ ψ β } be as in Example 3.7. Then F Q isinvariant under the first Markov move.Proof. Let Q = { Q β := ψ β ◦ γ β : F n ( β ) (cid:16) Γ ψ β } . Let n (cid:62) t >
0, and α, β ∈ B n . We will prove thatdet r Γ ψα − βα (cid:16) B (2) t,Q α − βα ( α − βα ) − Id ⊕ ( n − (cid:17) = det r Γ ψβ (cid:16) B (2) t,Q β ( β ) − Id ⊕ ( n − (cid:17) . Remark that, for any epimorphism γ : F n → G , it follows from Proposition 2.4that: Id ⊕ ( n − = B (2) t,γ (1) = B (2) t,γ ( αα − ) = B (2) t,γ ( α − ) ◦ B (2) t,γ ◦ h α − ( α ) , thus B (2) t,γ ( α − ) = (cid:16) B (2) t,γ ◦ h α − ( α ) (cid:17) − .Consequently, we have for any epimorphism γ : F n → G : B (2) t,γ ( α − βα ) = B (2) t,γ ( α ) ◦ B (2) t,γ ◦ h α ( β ) ◦ B (2) t,γ ◦ h βα ( α − )= B (2) t,γ ( α ) ◦ B (2) t,γ ◦ h α ( β ) ◦ (cid:16) B (2) t,γ ◦ h α − βα ( α ) (cid:17) − . Hence, for any epimorphism γ : F n → G :det rG (cid:16) B (2) t,γ ( α − βα ) − Id ⊕ ( n − (cid:17) = det rG (cid:18) B (2) t,γ ( α ) ◦ B (2) t,γ ◦ h α ( β ) ◦ (cid:16) B (2) t,γ ◦ h α − βα ( α ) (cid:17) − − Id ⊕ ( n − (cid:19) = det rG (cid:18) B (2) t,γ ◦ h α ( β ) − (cid:16) B (2) t,γ ( α ) (cid:17) − ◦ B (2) t,γ ◦ h α − βα ( α ) (cid:19) , where the second equality follows from Remark 2.5 and Proposition 2.2 (1).For γ = Q α − βα , we thus have:det r Γ ψα − βα (cid:16) B (2) t,Q α − βα ( α − βα ) − Id ⊕ ( n − (cid:17) = det r Γ ψα − βα (cid:18) B (2) t,Q α − βα ◦ h α ( β ) − (cid:16) B (2) t,Q α − βα ( α ) (cid:17) − ◦ B (2) t,Q α − βα ◦ h α − βα ( α ) (cid:19) = det r Γ ψα − βα (cid:18) B (2) t,χ Q β,α ◦ Q β ( β ) − (cid:16) B (2) t,Q α − βα ( α ) (cid:17) − ◦ B (2) t,Q α − βα ◦ h α − βα ( α ) (cid:19) . Now, since for every braid σ ∈ B n , γ σ ◦ h σ = γ σ and Q σ = ψ σ ◦ γ σ , we concludethat:det r Γ ψα − βα (cid:16) B (2) t,Q α − βα ( α − βα ) − Id ⊕ ( n − (cid:17) = det r Γ ψα − βα (cid:18) B (2) t,χ Q β,α ◦ Q β ( β ) − (cid:16) B (2) t,Q α − βα ( α ) (cid:17) − ◦ B (2) t,Q α − βα ◦ h α − βα ( α ) (cid:19) = det r Γ ψα − βα (cid:18) B (2) t,χ Q β,α ◦ Q β ( β ) − (cid:16) B (2) t,Q α − βα ( α ) (cid:17) − ◦ B (2) t,Q α − βα ( α ) (cid:19) = det r Γ ψα − βα (cid:16) B (2) t,χ Q β,α ◦ Q β ( β ) − Id ⊕ ( n − (cid:17) = det r Γ ψβ (cid:16) B (2) t,Q β ( β ) − Id ⊕ ( n − (cid:17) , where the last equality follows from Proposition 2.2 (3). (cid:3) Lemma 4.3.
Let Q = { ψ β ◦ γ β : F n ( β ) (cid:16) Γ ψ β } be as in Example 3.7. Then F Q isinvariant under the second Markov move.Proof. For any braid β let us denote Q β := ψ β ◦ γ β : F n ( β ) (cid:16) Γ ψ β .First step: negative crossing:Let n (cid:62) β ∈ B n and β − = σ − n ι ( β ) ∈ B n +1 . Let t >
0. We willprove thatdet r Γ ψβ − (cid:16) B (2) t,Q β − ( β − ) − Id ⊕ n (cid:17) max(1 , t ) n +1 = 1 t · det r Γ ψβ (cid:16) B (2) t,Q β ( β ) − Id ⊕ ( n − (cid:17) max(1 , t ) n . In order to do this, we will compose B (2) t,Q β − ( β − ) − Id ⊕ n with two operators (onenamed G on the left, one named D on the right) so that we obtain a block triangularoperator with the upper left block “mostly” equal to B (2) t,Q β ( β ) − Id ⊕ ( n − .First, it follows from Proposition 2.4 that B (2) t,Q β − ( β − ) = B (2) t,Q β − (cid:0) σ − n ι ( β ) (cid:1) = B (2) t,Q β − ( ι ( β )) B (2) t,Q β − ◦ h ι ( β ) (cid:0) σ − n (cid:1) . ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 11 Recall that B (2) t,id (cid:0) σ − n (cid:1) = Id 0. . . ...Id 0Id Id0 . . . − t R g n − g − n , thus the operatordefined as D := (cid:16) B (2) t,Q β − ◦ h ι ( β ) (cid:0) σ − n (cid:1)(cid:17) − is equal to: D := (cid:16) B (2) t,Q β − ◦ h ι ( β ) (cid:0) σ − n (cid:1)(cid:17) − = Id 0. . . ...Id 0Id tR ( Q β − ◦ h ι ( β ) ) ( g n g − n − ) . . . − tR ( Q β − ◦ h ι ( β ) ) ( g n g − n − ) . We therefore compute: B (2) t,Q β − ( β − ) ◦ D = B (2) t,Q β − ( ι ( β )) ◦ B (2) t,Q β − ◦ h ι ( β ) (cid:0) σ − n (cid:1) ◦ D = B (2) t,Q β − ( ι ( β ))= B (2) t,Q β − ◦ ι F n ( β ) ...0 R Id , where the row R is the right multiplication operator by the row (cid:18) κ ( t, Φ n +1 , Q β − ) (cid:18) ∂h ι ( β ) ( g j ) ∂g n (cid:19)(cid:19) = (cid:18) κ ( t, Φ n +1 ◦ ι F n , Q β − ◦ ι F n ) (cid:18) ∂h β ( g j ) ∂g n (cid:19)(cid:19) that has n − R Γ ψ β − .Now, the fundamental formula of Fox calculus (Proposition 2.1) implies that: (cid:0) R g − . . . R g n − (cid:1) · R ∂h ι ( β ) ( g j ) ∂g i ! (cid:54) i,j (cid:54) n = (cid:0) R h ι ( β ) ( g ) − . . . R h ι ( β ) ( g n ) − (cid:1) , thus, by applying κ ( t, Φ n +1 , Q β − ) to the previous equality and by definition of B (2) t,Q β − ( ι ( β )), we obtain: (cid:16) tR Q β − ( g ) − Id . . . t n R Q β − ( g n ) − Id (cid:17) · B (2) t,Q β − ( ι ( β ))= (cid:16) tR ( Q β − ◦ h ι ( β ) ) ( g ) − Id . . . t n R ( Q β − ◦ h ι ( β ) ) ( g n ) − Id (cid:17) = (cid:16) tR ( Q β − ◦ h ι ( β ) ) ( g ) − Id . . . t n − R ( Q β − ◦ h ι ( β ) ) ( g n − ) − Id t n R Q β − ( g n ) − Id (cid:17) , where the second equality follows from the fact that β ∈ B n leaves g n unchangedin the Artin action. Let us thus define the following block triangular operator G : G := Id 0. . . ...Id 0 tR Q β − ( g ) − Id . . . t n − R Q β − ( g n − ) − Id t n R Q β − ( g n ) − Id . It immediately follows from what precedes that G ◦ (cid:16) B (2) t,Q β − ( β − ) (cid:17) ◦ D = G ◦ (cid:16) B (2) t,Q β − ( ι ( β )) (cid:17) = B (2) t,Q β − ◦ ι F n ( β ) ...0 tR ( Q β − ◦ h ι ( β ) ) ( g ) − Id . . . t n − R ( Q β − ◦ h ι ( β ) ) ( g n − ) − Id t n R Q β − ( g n ) − Id . On the other hand, we compute G ◦ (cid:0) Id ⊕ n (cid:1) ◦ D = Id 0. . . ...0Id tR ( Q β − ◦ h ι ( β ) ) ( g n g − n − ) tR Q β − ( g ) − Id . . . t n − R Q β − ( g n − ) − Id ? , with ? = t n R Q β − ( g n h ι ( β ) ( g − n − ) g n − ) − t n +1 R Q β − ( g n h ι ( β ) ( g − n − ) g n ).Hence G ◦ (cid:16) B (2) t,Q β − ( β − ) − Id ⊕ n (cid:17) ◦ D = B (2) t,Q β − ◦ ι F n ( β ) − Id ⊕ ( n − ...0 − tR ( Q β − ◦ h ι ( β ) ) ( g n g − n − ) . . . t j R ( Q β − ◦ h ι ( β ) ) ( g j ) − t j R Q β − ( g j ) . . . (cid:3) , where j ∈ { , . . . , n − } and (cid:3) = t n R Q β − ( g n ) − Id − t n R Q β − ( g n h ι ( β ) ( g − n − ) g n − ) + t n +1 R Q β − ( g n h ι ( β ) ( g − n − ) g n ) . Now we use the fact that our epimorphism Q β − plunges deeper than the braidclosure group: indeed, for every j ∈ { , . . . , n − } , we have: Q β − ( g j ) = (cid:0) Q β − ◦ h β − (cid:1) ( g j ) = (cid:16) Q β − ◦ h ι ( β ) ◦ h σ − n (cid:17) ( g j ) = (cid:0) Q β − ◦ h ι ( β ) (cid:1) ( g j ) . Thus G ◦ (cid:16) B (2) t,Q β − ( β − ) − Id ⊕ n (cid:17) ◦ D is actually upper block triangular equal to: B (2) t,Q β − ◦ ι F n ( β ) − Id ⊕ ( n − ...0 − tR ( Q β − ◦ h ι ( β ) ) ( g n g − n − ) − Id + t n +1 R Q β − ( g n h ι ( β ) ( g − n − ) g n ) . By applying det r Γ ψβ − to the previous equality, it therefore follows from Proposition2.2 and Remark 2.5 thatmax(1 , t ) n · det r Γ ψβ − (cid:16) B (2) t,Q β − ( β − ) − Id ⊕ n (cid:17) · t = det r Γ ψβ − (cid:16) B (2) t,Q β − ◦ ι F n ( β ) − Id ⊕ ( n − (cid:17) · max(1 , t ) n +1 . We conclude by using the fact that Q β − ◦ ι F n = σ Q β, − ◦ Q β (by Markov-admissibilityof Q ) and Proposition 2.2 (3). ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 13 Second step: positive crossing:Let β + = σ n ι ( β ) ∈ B n +1 . We will proceed almost exactly as in the first step,except for the following differences: • We now aim to prove thatdet r Γ ψβ + (cid:16) B (2) t,Q β + ( β + ) − Id ⊕ n (cid:17) max(1 , t ) n +1 = det r Γ ψβ (cid:16) B (2) t,Q β ( β ) − Id ⊕ ( n − (cid:17) max(1 , t ) n . • The operator D becomes: D := (cid:16) B (2) t,Q β + ◦ h ι ( β ) ( σ n ) (cid:17) − = Id 0. . . ...Id 0Id Id0 . . . − t R Q β + ( g n g − n +1 ) . • The final lower right coefficient ? of G ◦ D becomes ? = t n − R Q β + ( g n − ) − Id − t n − R Q β + ( g n g − n +1 g n ) + 1 t R Q β + ( g n g − n +1 ) . • The final lower right coefficient (cid:3) of G ◦ (cid:16) B (2) t,Q β + ( β + ) − Id ⊕ n (cid:17) ◦ D becomes (cid:3) = − t n − R Q β + ( g n − ) + t n R Q β + ( g n ) + t n − R Q β + ( g n g − n +1 g n ) − t R Q β + ( g n g − n +1 ) . • The simplification (cid:3) = t n R Q β + ( g n ) − t R Q β + ( g n g − n +1 )comes from the following reasoning. Remark that in the ring ZF n +1 wehave the equalities: g n − − g n g − n +1 g n = g n (cid:0) g − n − g − n +1 g n g − n − (cid:1) g n − = g n (cid:0) g − n − h − σ n (cid:0) g − n (cid:1)(cid:1) g n − = g n (cid:16) g − n − h − σ n (cid:16) h − ι ( β ) ( g − n ) (cid:17)(cid:17) g n − = g n (cid:16) g − n − h − β + (cid:0) g − n (cid:1)(cid:17) g n − = g n (cid:16) h β + (cid:16) h − β + (cid:0) g − n (cid:1)(cid:17) − h − β + (cid:0) g − n (cid:1)(cid:17) g n − . Hence, by composing with γ β + : Z [ F n ] → Z [ G β + ], the quotient by all rela-tions of the form h β + ( ∗ ) = ∗ , we obtain γ β + (cid:0) g n − − g n g − n +1 g n (cid:1) = 0 . Theprevious equalities to zero remain true when applying any deeper epimor-phism Q β + = ψ β + ◦ γ β + instead. • Applying det r Γ ψβ + to the final equality now yieldsmax(1 , t ) n · det r Γ ψβ + (cid:16) B (2) t,Q β + ( β + ) − Id ⊕ n (cid:17) · t = det r Γ ψβ + (cid:16) B (2) t,Q β + ◦ ι F n ( β ) − Id ⊕ ( n − (cid:17) · t max(1 , t ) n +1 , as expected. (cid:3) We now have all the tools to prove Theorem 4.1.
Proof of Theorem 4.1.
This follows immediately from Lemmas 4.2 and 4.3. (cid:3) Counter-examples to Markov invariance
Theorem 4.1 established that F Q is a Markov function when the Markov-admissiblefamily of epimorphisms Q plunges to the link groups or deeper. It is now naturalto ask if those families Q are the only ones for which F Q is a Markov function. Un-fortunately, we do not have a clear answer to this question at the time of writing.However, by looking at the details of the proof of Theorem 4.1 (more preciselythose of Lemmas 4.2 and 4.3), it appears that applying the epimorphism γ β (ora strictly deeper epimorphism) is necessary in order to obtain the cancellations inthe matrices that yield Markov invariance for F Q . As additional evidence for thishypothesis, we discovered two families Q such that F Q is not a Markov function:one family (the identities of the free group) lives strictly higher than the γ β , andthe other one (the abelianizations of the free groups) is not comparable to the γ β (Figure 3 can help visualizing this). See the following Theorems 5.1 and 5.5.These two counter-examples illustrate some of the difficulties in computing Fuglede-Kadison determinants, and some transversal techniques one might need to use inorder to do so (techniques such as computation of Mahler measures of polynomials,or combinatorics of closed paths on Cayley graphs).Of course, appeareances might still deceive, and it could happen that unex-pected identities of Fuglede-Kadison determinants occur even with families of epi-morphisms not encompassed by Theorem 4.1, especially since computing such de-terminants remains a daunting task to the day of writing.5.1. Descending to the free abelian group.
In the following theorem, we provethat when the family of epimorphisms Q descends to the free abelian groups, F Q is not a Markov function and thus cannot yield knot invariants. The proof usesproperties of the Fuglede-Kadison determinant and a value of the Mahler measuredue to Boyd [Bo]. Theorem 5.1.
For the family of abelianizations Q = (cid:8) ϕ n ( β ) : F n ( β ) (cid:16) Z n ( β ) (cid:9) , thevalue at t = 1 of the function F Q is not invariant under Markov moves. In particular F Q is not a Markov function.Proof. Let t > , n = 2 , β = σ − ∈ B , and β + = σ − σ ∈ B . Following Section2.1, we compute that β + acts on F by (cid:26) g g h β + (cid:26) g − g g g − g − g , where g , g , g denote the generators as in Section 2.1. Thus Fox calculus gives: B (2) t,id ( β + ) − Id ⊕ = − t R g − − Id − R g g − g − t R g − − Id − tR g g − + R g g − g − ! . Hence, from Remark 2.3 and Proposition 2.2 (6), we have:det F (cid:16) B (2) t,id ( β + ) − Id ⊕ (cid:17) = det F (cid:18)(cid:16) − Id − tR g g − + R g g − g − (cid:17) (cid:16) − R g g g − (cid:17) (cid:18) − t R g − − Id (cid:19) − t R g − (cid:19) = det F (cid:18) − (cid:18) tR g + R g g g − + 1 t R g g − (cid:19)(cid:19) = 1 t det F (cid:16) Id + tR g + t R g g g − (cid:17) . ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 15 Let z , z , z denote the canonical generators of Z . Then it follows from thesame arguments as in the previous computation that:det F (cid:16) B (2) t,ϕ ( β + ) − Id ⊕ (cid:17) = 1 t det Z (cid:0) Id + tR z + t R z z (cid:1) . Let H be the subgroup of Z isomorphic to Z and generated by λ = z and µ = z z . Recall that F Q ( β + ) is an equivalence class of functions of t > t = 1 is always the same regardlessof the representant of the equivalence class. Let us denote this value F Q ( β + )(1).We then have: F Q ( β + )(1) = det Z (Id + R z + R z z )= det H (Id + R λ + R µ )= M (1 + X + Y ) = 1 . ... = 1 . where M is the Mahler measure of a polynomial, the second equality follows fromProposition 2.2 (3), the third one from Proposition 2.2 (7) and the fourth one from[Bo, Section 4].Now, since by Proposition 2.2 (5) we have: F Q ( β )(1) = det Z (cid:16) − R ϕ ( g − g ) − Id (cid:17) = 1 , we conclude that β F Q ( β )(1) is not invariant under Markov moves. (cid:3) Remaining at the free group.
The following lemma gives a method tocompute Fuglede-Kadison determinants using the generating series associated tothe number of closed paths on a regular graph (see [Ba, DL] and [Lu, Section 3.7]).
Lemma 5.2.
Let G be a finitely presented group and A ∈ R C G the right multipli-cation operator on ‘ ( G ) by a non-zero element of the group algebra. Then for any λ ∈ (cid:0) , k A k − (cid:1) , we have: det G ( A ) = lim ε → + √ λ exp (cid:18) − Z w λ,ε ( t ) − t dt (cid:19) , where w λ,ε ( t ) = ∞ X n =0 tr G (((1 − λε )Id − λA ∗ A ) n ) t n is a well-defined power series for ε small enough and t < .Proof. Let λ ∈ (cid:0) , k A k − (cid:1) . Since A is assumed to be non-zero, it is injective byRemark 2.3. Thus it follows from Proposition 2.2 (8) thatdet G ( A ) = lim ε → + p det G ( A ∗ A + ε Id) . Let ε > λ < k A k + ε < k A k . Since A ∗ A + ε Id is a positive operator,we have:det G ( A ∗ A + ε Id) = (exp ◦ tr G ◦ ln) ( A ∗ A + ε Id)= 1 λ (exp ◦ tr G ◦ ln) ( λA ∗ A + λε Id)= 1 λ (exp ◦ tr G ◦ ln) (Id − (Id − ( λA ∗ A + λε Id)))= 1 λ (exp ◦ tr G ) − ∞ X n =1 n (Id − ( λA ∗ A + λε Id)) n ! = 1 λ exp − ∞ X n =1 n tr G (((1 − λε )Id − λA ∗ A ) n ) ! , where the first equality follows from Proposition 2.2 (4), the fourth one from holo-morphic functional calculus and the fact that the spectrum of the positive operator λA ∗ A + λε Id is inside (0 ,
1) (since λ < k A k + ε ).Now, since the series P ∞ n =1 1 n tr G (((1 − λε )Id − λA ∗ A ) n ) converges, we thereforehave that w λ,ε ( t ) := ∞ X n =0 tr G (((1 − λε )Id − λA ∗ A ) n ) t n is a well-defined power series for t <
1, and moreover that for all
T < ∞ X n =1 n tr G (((1 − λε )Id − λA ∗ A ) n ) T n = Z T w λ,ε ( t ) − t dt. Finally, once again since P ∞ n =1 1 n tr G (((1 − λε )Id − λA ∗ A ) n ) converges, we can ap-ply Abel’s theorem and make T → − in the previous equality. Hence:det G ( A ) = lim ε → + p det G ( A ∗ A + ε Id)= lim ε → + vuut λ exp − ∞ X n =1 n tr G (((1 − λε )Id − λA ∗ A ) n ) ! = lim ε → + s λ exp (cid:18) − Z w λ,ε ( t ) − t dt (cid:19) , and the lemma follows. (cid:3) In the following proposition, we compute Fuglede-Kadison determinants of basicoperators on the free groups, using the explicit value (given in [Ba, DL]) of thegenerating series counting paths on a regular infinite tree.
Proposition 5.3.
Let d (cid:62) , and let x , . . . , x d − be d − generators of the freegroup F d − . Then we have: det F d − (Id + R x + . . . + R x d − ) = ( d − d − d d − . In particular, for any two generators x, y of the free group F , we have: det F (Id + R x + R y ) = 2 √ . ... The following proof is somewhat technical but only uses classical calculus argu-ments.
ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 17 Proof.
Let d (cid:62)
3. First we use Lemma 5.2, by taking G = F d − , A = Id + R x + . . . + R x d − , λ ∈ (0 , d ), and we obtain:det F d − (Id+ R x + . . . + R x d − ) = det G ( A ) = lim ε → + √ λ exp (cid:18) − Z w λ,ε ( t ) − t dt (cid:19) , where w λ,ε ( t ) = ∞ X n =0 tr G (((1 − λε )Id − λA ∗ A ) n ) t n for ε > t ∈ [0 , ε > t ∈ [0 , w λ,ε ( t ) = ∞ X n =0 tr G (((1 − λε )Id − λA ∗ A ) n ) t n = ∞ X n =0 tr G n X k =0 (cid:18) nk (cid:19) (1 − λε ) n − k ( − λ ) k ( A ∗ A ) k ! t n = ∞ X n =0 n X k =0 (cid:18) nk (cid:19) (1 − λε ) n − k ( − λ ) k tr G (cid:0) ( A ∗ A ) k (cid:1) t n = ∞ X k =0 (cid:18) − λ − λε (cid:19) k tr G (cid:0) ( A ∗ A ) k (cid:1) ∞ X n = k (cid:18) nk (cid:19) (1 − λε ) n t n = ∞ X k =0 (cid:18) − λ − λε (cid:19) k tr G (cid:0) ( A ∗ A ) k (cid:1) ((1 − λε ) t ) k (1 − (1 − λε ) t ) k +1 = 11 − (1 − λε ) t ∞ X k =0 (cid:18) − λt − (1 − λε ) t (cid:19) k tr G (cid:0) ( A ∗ A ) k (cid:1) , where the fifth equality follows from the binomial formula. Thus, by denoting u ( t ) := P ∞ k =0 tr G (cid:0) ( A ∗ A ) k (cid:1) t k , we have w λ,ε ( t ) = 11 − (1 − λε ) t u (cid:18) − λt − (1 − λε ) t (cid:19) . Since it suffices to prove thatlim ε → + Z w λ,ε ( t ) − t dt = ln (cid:18) d d − ( d − d − λ (cid:19) , we will study the integral I λ,ε := R w λ,ε ( t ) − t dt .It follows from [DL] and [Ba] that u ( t ) = 2 d − d − d p − d − t , thus I λ,ε = Z
10 11 − (1 − λε ) t d − d − d q d − λt − (1 − λε ) t − t dt. An antiderivative of the integrand is F ( t ) := ( d −
2) ln (cid:18) − (cid:18) − d λ − λε (cid:19) (1 − λε ) t (cid:19) + d − (1 − (1 − λε ) t ) q d − λt − (1 − λε ) t + (cid:16) d − λ − λε − (cid:17) (1 − λε ) t (1 − λε ) t + (cid:18) − d (cid:19) ln ( − d ( d − − (1 − λε ) t ) s d − λt − (1 − λε ) t − ! d ( d − λt + 2(1 − (1 − λε ) t ) ) . It remains to compute I λ,ε = F (1) − F (0). At t = 0, we first remark that: s d − λt − (1 − λε ) t = t → + d − λt − (1 − λε ) t − d − λ t (1 − (1 − λε ) t ) + o t → + ( t ) , thus, from an immediate calculation, the fraction inside the ln in the second termof F ( t ) above behaves like 2 λ ( d − − (1 − λε ) t + o t → + (1) . Hence F (0) = ( d −
2) ln(1) + d (cid:0) λ ( d − (cid:1) + (cid:18) − d (cid:19) ln(2) = ln (cid:0) λ d ( d − d (cid:1) . Furthermore, at t = 1 we compute: F (1) = ( d −
2) ln( d λ + λε )+ d (cid:18) λ (1 − λε ) (cid:16) d −
1) + ε − √ ε p ε + 4( d − (cid:17)(cid:19) + (cid:18) − d (cid:19) ln (cid:16) d ( d − λ + 2 λε − d ( d − λ √ ε (cid:16)p ε + 4( d − − √ ε (cid:17)(cid:17) . Taking ε → + , we find: F (1) −→ ε → + ( d −
2) ln( d λ ) + d d − λ ) + (cid:18) − d (cid:19) ln (cid:0) d ( d − λ (cid:1) = ln (cid:0) λ d − d d − ( d − (cid:1) . Hence we obtain the limit I λ,ε = F (1) − F (0) −→ ε → + ln (cid:0) λ − d d − ( d − − d (cid:1) , and the lemma follows. (cid:3) Remark . In [Lu2], W. L¨uck defined Lehmer’s constant Λ w ( G ) for any group G , as the infimum of Fuglede-Kadison determinants det G ( A ) of injective operators A ∈ R Z G such that det G ( A ) > F d (for d (cid:62)
2) injects in F and vice-versa, it followsfrom Proposition 2.2 (3) that all free groups F d have the same Lehmer’s constantΛ w ( F d ).Furthermore, Proposition 5.3 provides us a new upper bound on this constant: ∀ d (cid:62) , Λ w ( F d ) (cid:54) √ . ... ARKOV MOVES, L -BURAU MAPS AND LINK INVARIANTS 19 Indeed, the operators Id + R x + . . . + R x d − are injective because the free groupssatisfy the Strong Atiyah Conjecture (see Remark 2.3 and [Lu, Theorem 10.19]).We can now prove that we cannot expect Markov invariance by remaining at thelevel of the free group, as the following theorem shows: Theorem 5.5.
For the family of identities Q = { id F n ( β ) } , the function F Q is notinvariant under Markov moves.Proof. Let us take t = 1 , n = 2 , β = σ − ∈ B , β + = σ − σ ∈ B . Then, as inthe proof of Theorem 5.1, we obtain F Q ( β + )(1) = det F (cid:16) B (2)1 ,id ( β + ) − Id ⊕ (cid:17) = det F (cid:16) Id + R g + R g g g − (cid:17) = det F (Id + R x + R y ) , where F is the free group on two generators x, y that embeds in F via x g , y g g g − , and the last equality follows from Proposition 2.2 (3). Now, since F Q ( β )(1) = det F (cid:16) − R g − g − Id (cid:17) = 1 , it then suffices to prove that det F (Id + R x + R y ) = 1. This is the case thanks toProposition 5.3. (cid:3) References [Ba] L. Bartholdi,
Counting paths in graphs , Enseign. Math. (2) 45 (1999), no. 1-2, 83–131.[BAC] F. Ben Aribi and A. Conway, L -Burau maps and L -Alexander torsions (2018), OsakaJ.Math. 55, 529-545.[Bo] D. Boyd, Speculations concerning the range of Mahler’s measure , Canad. Math. Bull. 24(1981), no. 4, 453––469.[DL] O. Dasbach and M. Lalin,
Mahler measure under variations of the base group , Forum Math.21 (2009), no. 4, 621–637.[DFL] J. Dubois, S. Friedl and W. L¨uck,
The L -Alexander torsion of 3-manifolds , Journal ofTopology 9, No. 3 (2016), 889–926.[Fox] R.H. Fox, Free differential calculus. II. The isomorphism problem of groups , Ann. of Math.(2), 59 (1954), 196–210.[LZ] W. Li and W. Zhang, An L -Alexander invariant for knots , Commun. Contemp. Math. (2006), no. 2, 167–187.[Lu] W. L¨uck, L -invariants: theory and applications to geometry and K -theory , Ergebnisse derMathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,44. Springer-Verlag, Berlin, 2002.[Lu2] W. L¨uck, Lehmer’s Problem for arbitrary groups , to appear in Journal of Topology andAnalysis, arXiv:1901.00827.
UCLouvain, IRMP, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Email address ::