Twisted Alexander invariants of knot group representations II; computation and duality
aa r X i v : . [ m a t h . G T ] M a y Twisted Alexander invariants of knot group representations II;computation and duality .Takefumi Nosaka Abstract
Given a homomorphism from a link group to a group, we introduce a K -class, which is a generalizationof the 1-variable Alexander polynomial. We compare the K -class with K -classes in [Nos] and withReidemeister torsions. As a corollary, we show a relation to Reidemeister torsions of finite cyclic coveringspaces, and show reciprocity in some senses. Keywords knot, Alexander polynomial, K -group, Novikov ring Let L ⊂ S be a knot in the 3-sphere. The Alexander polynomial of L has been studied inmany ways, and its applications and beautiful properties are discovered, including, e.g., rela-tions to Reidemeister torsions, Fox derivatives, and applications to cyclic coverings and sliceknots. As a generalization, after twisted Alexander polynomials are introduced [Wada, Lin],similar applications and properties are discovered (however, such results frequently requiresome assumptions); see [FV, FKK] and references therein.In the previous paper [Nos], which is inspired by [Mil1, Tur], given a homomorphism from alink group to a group,the author suggested elements of a K -group, which are a generalizationof the twisted Alexander polynomials. This paper gives several approaches to the K -classin another way. In Sections 2 and 4, we give two other definitions as K -classes from theviewpoint of the Fox derivatives or Reidemeister torsion over K . Here, an advantage of thedefinitions is applicable to not only knots but also links; see Definition 2.2. Furthermore,in the knot case, we show the equivalence (up to some ambiguity) of the three definitions(Theorem 3.1). As a corollary of the equivalence, we compute the K -classes of some 2-bridgeknots; see Section 2.1.In Sections 5–7, we will address some applications. First, under some conditions, we give arelation to (commutative) Reidemeister torsion of the m -fold cyclic covering space of S \ L ;see Section 5. Next, we will see that the conditions are suitable to group homomorphisms f : π ( S \ L ) → Z ⋊ G , where G is a finite group. In this situation, we show (Theorem 6.1)a duality theorem of the K -class, as in reciprocity of the (twisted) Alexander polynomials[FKK, Mil1, Tur]. In application, we give an estimate of sliceness of knots, which is a slightgeneralization of the works of Herald-Kirk-Livingston [HKL, KL]. In fact, we find that theconditions in Section 7 are applicable to the Casson-Gordan theory [CG].Finally, under some conditions, we give a relation to the circle valued Morse theory (seeAppendix A), and a reduction to the higher order Alexander polynomial [C, Har]; see AppendixB. These discussions rely on the works of [P, Fri2]. E-mail address: [email protected] onventional notation. For a group G and a commutative ring A , we denote the groupring by A [ G ], and the abelianization by G ab . Every non-commutative ring R has always 1,and is assumed to satisfy that R r and R s with r = s are not isomorphic as R -modules. Wemean by R × the multiplicative group consisting of units in R . Acknowledgments
The author expresses his gratitude to Takahiro Kitayama, Ryoto Tange for valuable comments. K -class We give the definition as a generalization of the Alexander polynomial; see Definition 2.2.As a similar setting to [Nos], we set up algebraic terminologies and need an assumption.Let A be a ring, which is possibly non-commutative, and take a ring isomorphism κ : A → A .Then, we have the completed skew Laurent-polynomial ring A κ (( τ )). Namely, A κ (( τ )) is the setof formal power series P ∞ i = − N a i τ i where a i ∈ A and τ n a = κ n ( a ) τ n ; in other words, A κ (( τ ))is equal to A κ [[ τ ]][ τ − ], which is also called the Novikov ring in [Fri1, PR]. Furthermore, letus fix a group G with presentation h x , . . . , x m | r , . . . , r m − i of deficiency 1, and consider thefollowing assumption throughout this paper. Assumption ( † ) Let A be a ring and κ : A → A be as above. Suppose a ring homomorphism ρ : Z [ G ] → A κ (( τ )) such that, for any i ≤ m , there is w i ∈ A κ (( τ )) × such taht ρ ( x i ) = w − i τ w i . Example 2.1 (A special case of Novikov completion) . Set a link L in the 3-sphere S . Choosea link diagram D , and let m be the number of the arcs on D . Then, the Wirtinger presentationfrom D gives such a presentation π ( S \ L ) ∼ = h x , . . . , x m | r , . . . , r m − i , possibly r j = 1 (see,e.g., [Lic], [Wada, § h : π ( S \ L ) → H ⋊Z such that H is a group, h ( x i ) = ( g i , g i ∈ G . Let A be the group ring B [ H ] over a commutative ring B . If we replace h ( x i )by h ( x i ) τ and define κ ( a ) := (0 , a, , −
1) for a ∈ H we have A κ (( τ )). This h canonicallygives rise to ρ : Z [ π ( S \ L )] → A κ (( τ )) satisfying ( † ) . In general, we obtain such an h from any group G and any group homomorphism f : π ( S \ L ) → G , as follows. Let Ab : π ( S \ L ) → Z ♯L be an abelianization, and µ : Z ♯L → Z be the multiplication such that µ ◦ Ab( x i ) = 1 for any i . Notice the isomorphism π ( S \ L ) ∼ =Ker( µ ◦ Ab) ⋊ Z . Let H ⊂ G be the restricted image f (Ker( µ ◦ Ab)), and let Z = { τ n } n ∈ Z act on H by g · τ n := f ( x ) n gf ( x ) − n . By the action, we have the semi-direct product H ⋊ Z ,and can define a homomorphism h : π ( S \ L ) = Ker( µ ◦ Ab) ⋊ Z → H ⋊ Z by h ( g, n ) = ( f ( g ) , n ) , which satisfy the condition in the previous paragraph.Let us review up K -groups. For a ring R with unit, let GL n ( R ) be the general lineargroup over R of size n . Since GL n ( R ) diagonally injects into GL n +1 ( R ), we have the colimit GL ( R ) = lim GL n ( R ). The K -group, K ( R ), is defined to be the abelianization GL ( R ) ab . Byabuse of notation, we often regard elements of GL n ( R ) as those of K ( R ). When R = A κ (( τ )) s above, − τ , and ρ ( g ) are represented by the invertible (1 × ± τ denote thesubgroup of K ( A κ (( τ ))) generated by − τ , which is isomorphic to either Z or Z × Z / ± ρ ( G ) be the subgroup of K ( A κ (( τ ))) generated by − ρ ( G ). Welater use the quotient groups K ( A κ (( τ ))) / ± τ and K ( A κ (( τ ))) / ± ρ ( G ).Furthermore, we study the Fox derivative and a Jocobi matrix. Let F be the free group witha basis, x , . . . , x m . We define a Z -linear map ∂∂x i : Z [ F ] → Z [ F ] by the following identities: ∂x j ∂x i = δ ij , ∂ ( hk ) ∂x i = ∂h∂x i + h ∂k∂x i ( h, k ∈ F ) . Consider the ( m − × m matrix, A ρ , over A κ (( τ )) whose ( i, j )-th entry is ρ ( ∂r i ∂x j ) ∈ A κ (( τ )).For 1 ≤ k ≤ m , let us denote by A ρ,k by the ( m − × ( m − A ρ byremoving the k -th column. Definition 2.2.
Choose k ∈ N as above. Suppose that A ρ,k is an invertible matrix. We define the K -Alexander class (with respect to ρ ) to be the K -class[ A ρ,k ] ∈ K ( A κ (( τ ))) / ± τ. On the other hand, if A ρ,k is not invertible, we define the K -class to be zero. Theorem 2.3.
Let G = π ( S \ L ) be a link group with a Wirtinger presentation, as in Example2.1. Then, the K -class ∆ K ρ depends only on the homomorphism ρ : Z [ π ( S \ L )] → A κ (( τ )) . We will give the proof in § A F,W , we give a criterionfor the invertibility:
Proposition 2.4.
Suppose ♯L = 1 , i.e., L is a knot. Then, L is fibered if and only if A ρ,k isinvertible for any homomorphism ρ satisfying ( † ). We conclude this section by explaining that the twisted Alexander polynomial of [Wada]can be formulated from our K -Alexander class: Example 2.5.
Let R be a commutative ring. The paper [Wada, §
5] defines a polynomial froma representation ρ pre : π ( S \ L ) → SL n ( R ) as follows. For the formulation, let A be the matrixring Mat( n × n, R ), and let κ be the identity id A . Formally, let τ be ρ pre ( m ) as a commutativeindeterminate. Since there is a ring homomorphism Z [ SL n ( R )] → Mat( n × n, R ) which sends P g a g g to P g a g g , the ρ pre gives rise to a ring homomorphism ρ : Z [ π ( S \ L )] → A κ (( τ )).Notice that the determinant Mat( n × n, R κ (( τ ))) → R κ (( τ )) induces a homomorphismdet : K ( A κ (( τ ))) / ± τ −→ R κ (( τ )) × / ± τ. Then, if L = 1, checking [Wada, Corollary 5] carefully, we verify by construction that thedeterminant det(∆ K ρ ) ∈ R κ (( τ )) × / ± τ exactly coincides with the twisted Alexander polynomialof [Wada]. In other words, our K -class ∆ K ρ is a lift of the twisted Alexander polynomial. We will compute the K -classes with respect to every 2-bridge knots of Seifert genus 1. - t w i s t m -twist Figure 1: The 2-bridge knot of genus 1.
For non-zero integers m and n , let K ( m, n ) be the 2-bridge knot S (4 mn + 1 , m ) in Schu-bert’s form; see Figure 1. We may suppose m >
0. As is known, every 2-bridge knot ofgenus 1 is represented as one of K ( m, n ). Thanks to Proposition 2.1 in [HT], the knot group π ( S \ K ( m, n )) has the presentation h x, y | w n x = yw n i , where w = ( xy − ) m ( x − y ) m , where x and y are conjugate to a meridian. Moreover, as seen in the proof of the proposition,this presentation is strongly Tietze equivalent to a Wirtinger presentation. Proposition 2.6.
Let m > . Consider the following element of A κ (( τ )) : Z m,n := ( ρ ( w n ) + (1 − ρ ( y )) − ρ ( w n )1 − ρ ( w ) (cid:0) − ρ ( wx − ) (cid:1) − ρ ( xy − ) m − ρ ( xy − ) ( n > ,ρ ( w n ) + ( ρ ( y ) − ρ ( w − ) − ρ ( w n )1 − ρ ( w ) (cid:0) − ρ ( wx − ) (cid:1) − ρ ( xy − ) m − ρ ( xy − ) ( n < . The matrix A F,W is invertible if and only if Z m,n is an invertible element. If so, the K -Alexander invariant is given by Φ ρ,k = Z m,n ∈ K ( A κ (( τ ))) / ± τ. Proof.
Let r = w n xw − n y − . It is sufficient to show Z m,n = ρ ( ∂r∂x ). Notice that ∂r∂x = ∂w n ∂x + w n ∂∂x ( xw − n y − ) = ∂w n ∂x + w n (1 + x ∂∂x ( w − n ))= ( (1 + w + · · · + w n − ) ∂w∂x + w n + yw n (1 + w − + · · · + w − n ) ∂w − ∂x ( n > , (1 + w − + · · · + w n +1 ) ∂w∂x + w n + yw n (1 + w − + · · · + w − n − ) ∂w − ∂x ( n < , = ( w n + (1 − y )(1 + w + · · · + w n − ) ∂w∂x ( n > ,w n + ( y − w − + · · · + w n ) ∂w∂x ( n < . Here, the last equality is due to ∂w − ∂x = − w − ∂w∂x . We should observe ∂w∂x = (1 + xy − + · · · + ( xy − ) m − ) − ( xy − ) m (1 + x − y + · · · + ( x − y ) m − ) x − = (1 − ( xy − ) m y − ( yx − ) m )(1 + xy − + · · · + ( xy − ) m − )= (1 − wx − )(1 − ( xy − ) m ) / (1 − xy − ) . Therefore, we have Z m,n = ρ ( ∂r∂x ) as required.The above computation is done as an element of A κ (( τ ))); however, there are many examplessuch that Φ ρ,k the K -Alexander invariant Φ ρ,k can not be represented by any 1 × L is a non-fibered pretzel knot. Indeed, even concerning the classical Alexander module,the minimal number of sizes of the matrix Φ ρ,k is estimated by Nakanishi index. .2 Proof of Theorem 2.3 For the proof, we review the strongly Tietze transformations [Wada]. For a finite presentablegroup G = h g , . . . , g m | r , . . . , r n i and a word w of g , . . . , g m , the transformation of the fol-lowing types are called the strongly Tietze transformations :(Ia) To replace one of the relators r i by its inverse r − i . (Ib) To replace one of the relators r i by its conjugate wr − i w − . (Ic) To replace one of the relators r i by r i r j for any j = i .(II) To add a new generator y and a new relator yw − . In other words, the resultingpresentation is given by h g , . . . , g m , y | r , . . . , r n , yw − i .Two presentations of G are said to be strongly Tietze equivalent , if they are related by afinite sequence of the operations of the above types and their inverse operations. Proof of Theorem 2.3.
It is shown [Wada, Lemma 6 in §
5] that any two Wirtinger presenta-tions for a given link L are strongly Tietze equivalent. Hence, for the proof of Theorem 2.3,it is enough to show the following proposition, which claims an invariance with respect tostrongly Tietze equivalence: Proposition 2.7 (cf. [Ki1, Theorem 4.5]) . The K -class [Φ ρ,k ] in K ( A κ (( τ ))) / ± τ does notdepend on the choice of k . Moreover, if we change another presentation of G which is stronglyTietz equivalent to the above presentation, the associated K -class [Φ ρ,k ] in K ( A κ (( τ ))) / ± τ is invariant.Proof. The proof is essentially based on the proofs of [Wada, Lemma 2 in §
3] and [Ki1, Theorem4.5]. Following the proofs, we will check ( i ) the independence of the choice of k , and ( ii ) theinvariance with respect to each transformations (Ia)(Ib)(Ic)(II).To show ( i ), choose k ′ . Let e j ( a ) be the diagonal ( m − -matrix whose ( j, j )-th entry is a and ( k, k )-th entry is 1 where j = k . Then, according to the fundamental formula m X j =1 ∂r i ∂x j ( x j −
1) = r i − A ρ,k ′ e k ( ρ ( x k ) −
1) = (cid:16) . . . , ρ (cid:16) ∂r i ∂x k (cid:17) ρ ( x k − , . . . (cid:17) = (cid:16) . . . , − X j = k ρ (cid:16) ∂r i ∂x j (cid:17) ρ ( x j − , . . . (cid:17) = (cid:16) . . . , − ρ (cid:16) ∂r i ∂x k ′ (cid:17) ρ ( x k ′ − , . . . (cid:17) = ( − k − k ′ A ρ,k e k ′ ( ρ ( x k ′ ) − ∈ Mat( m × m ; A κ (( τ ))) . Notice that e k ( ρ ( x k ) −
1) is invertible, since 1 − ρ ( x k ) is invertible because of (cid:0) − ρ ( x k ) (cid:1) (1 + P ∞ j =1 w − k τ j w k ) = 1. Therefore, the invertibility of A ρ,k implies that A ρ,k ′ is also invertible.Moreover, since ρ ( x k ) − ρ ( w − k )( τ − ρ ( w k ) = τ − K by Assumption ( † ), we obtain A ρ,k ′ = A ρ,k in the quotient K ( A κ (( τ ))) / ± τ .Next, concerning ( ii ), we consider strongly Tietz transformations. Notice ∂ ( r − i ) ∂x j = − r − i ∂r i ∂x j , ∂ ( wr i w − ) ∂x j = w ∂r i ∂x j , ∂ ( r i r ℓ ) ∂x j = ∂r i ∂x j + r i ∂r ℓ ∂x j . herefore, the changes of the K -classes with respect to Ia and Ib are A ρ,k
7→ − A ρ,k , and A ρ,k ( − k ρ ( w ) A ρ,k , respectively. Furthermore, recalling from Whitehead Lemma [Wei,Lemma III.1.3.3] that any elementary matrix is 1 in K , we can easily verify that the changes ofthe K -values with respect to Ic and II are A ρ,k A ρ,k , and A ρ,k ρ ( w − ) A ρ,k , respectively.Since ρ ( x k ) = τ ∈ K ( A κ (( τ ))), we have ρ ( w ) = τ n w ∈ K ( A κ (( τ ))) for some n w ∈ Z byassumption. Hence, the observation in the quotient K ( A κ (( τ ))) / ± τ proves (ii). K -class in [Nos] In what follows, we let L be a knot embedded in the 3-sphere S , i.e., L = 1 , and choose aknot diagram on R .In the paper [Nos], the author constructed another K -class, where the construction isinspired by [Lin]. This section shows (Theorem 3.1) that this class and another [Φ ρ,k ] in § L is a knot, and choose a meridian m ∈ π ( S \ L )such that ρ ( m ) = τ .We recall the presentation (1) and the definition of the K -class below. We choose a Seifertsurface F of genus g and a bouquet of circles W ⊂ F such that W is a deformation retract of F and the inclusion F ⊂ S is isotopic to the standard embedding W ⊂ F . Take a bicollar F × [ − ,
1] of F such that F × { } = F . Let ι ± : F → S \ F be the embeddings whoseimages are F × {± } . Take generating sets W := { u , . . . , u g } of π F and X := { x , . . . , x g } of π ( S \ F ), and set y ♯i := ( ι + ) ∗ ( u i ) and z i = ( ι − ) ∗ ( u i ); a von Kampen argument yields apresentation h x , . . . , x g , m | r Fi := m − y i m z − i (1 ≤ i ≤ g ) i , (1)of π ( S \ L ). Notice that ∂r Fi ∂x j = m ∂y i ∂x j − r Fi ∂z i ∂x j , and consider the square matrix of the form A F,W := n ρ ( ∂r Fi ∂x j ) o ≤ i,j ≤ g = n τ ρ ( ∂y j ∂x i ) − ρ ( ∂z j ∂x i ) o ≤ i,j ≤ g ∈ Mat(2 g × g, A κ (( τ ))) . (2)Then, the paper [Nos, §
3] defined a quotient group Q A ,κ := K ( A κ (( τ ))) /K ( A ) , (3)and if A F,W is an invertible matrix, we define the K -Alexander invariant (with respect to ρ ) to be the K -class ∆ K ρ := [ τ − g A F,W ] ∈ Q A ,κ . On the other hand, if A F,W is not invertible, we define ∆ K ρ to be zero. The main theorem in[Nos] shows that this ∆ K ρ does not depend on the choice of F, W, and X .Then, the theorem of this section is as follows: Theorem 3.1.
Let G be a knot group with a Wirtinger presentation, as in Example 2.1, andlet ρ : Z [ π ( S \ L )] → A κ (( τ )) satisfy ( † ).Then, Φ ρ,k is invertible if and only if so is A F,W . In addition, if so, the K -class [ τ − g A F,W ] is equal to [Φ ρ,k ] in the quotient group Q A ,κ / { τ n } n ∈ Z . Review of the Reidemeister torsion in a K -group For the proof of Theorem 3.1, the Reidemeister torsion appearing in a K -group plays a keyrule. We will review the Reidemeister torsion. The definition and properties are based on[Mil1] or [Tur, § I.3].Consider an exact sequence of length 3 C ∗ : 0 → C ∂ −→ C ∂ −→ C ∂ −→ C → , where C ∗ is a finitely generated free R -module ( C may be zero). Let us choose a basis C i ⊂ C i for all i with C i = 0. Assume that B i = Im( ∂ i +1 ) ⊂ C i is free, pick a basis B i of B i and alift ˜ B i of B i to C i . By B i ˜ B i − we mean the collection of elements given by B i and ˜ B i − . Since C ∗ is exact, B i ˜ B i − is indeed a basis for C i . For bases d, e of a complex, denote by [ d/e ] theinvertible matrix of a basis change, i.e., [ d/e ] = ( a ij ) where d i = P j a ij e j . Then we define theReidemeister torsion of the based acyclic complex ( C ∗ , C i ) to be T ( C ∗ , C i ) := [ ˜ B / C ][ B ˜ B / C ] − [ B ˜ B / C ][ B / C ] − ∈ K ( R ) . Meanwhile, in the case where the R -modules B i are not free, Section 3 in [Tur] gives a definitionof the torsion of C ∗ . Since this paper does not consider such a case in details, we omit thedetails.We will study homology groups in local coefficients. Let X be a connected CW complex.Denote the universal covering space of X by ˜ X . We regard the chain complex of space, C ∗ ( ˜ X ), as a chain complex of right Z [ π ( X )]-modules, where the Z [ π ( X )]-module structureis defined via covering transformations. Given a ring homomorphism ρ : Z [ π ( X )] → R ,we can therefore consider the chain complex C ∗ ( X ; R ) = C ∗ ( ˜ X ) ⊗ Z [ π ( X )] R . We denote itshomology by H ∗ ( X ; R ).We now suppose that X is of finite type and dim X ≤
3. Then, if the homology H i ( X ; R )is not zero for some i , we write T ( X, ρ ) = 0 . Otherwise, denote the i -cells of X by σ i , . . . , σ r i i ,and choose an orientation for each cell σ ji , and also pick a lift ˜ σ ji for each cell σ ji to the universalcover e X . Since the set { ˜ σ i , . . . , ˜ σ r i i } makes a basis C i of C i ( X ; R ), we can define T ( C i ( X ; R ) , {C i } ) ∈ K ( R ) . (4)Moreover, consider the quotient of K ( R ) passage by the image {± ρ ( g ) } g ∈ π ( X ) . Then, as isknown, the class T ( X, ρ ) := [ T ( C i ( X ; R ) , {C i } )] ∈ K ( R ) / ± ρ ( π ( X )) (5)depends only on the simple homotopy type of X and the homomorphism ρ : π ( X ) → R .Next, we give a procedure for computing T ( X, ρ ), following [Tur, p. 8]. Take subsets, ξ i ⊂ { , , . . . , rank( C i ) } so that ξ = ∅ , and denote ( ξ , ξ , . . . , ξ m ) by ξ . For i, j, k , define thematrix { a ijk } j,k by considering only the ξ i -columns of A i and with the ξ i − -rows removed. Sucha matrix chain ξ is called a T -chain if A ( ξ ) , . . . , A m ( ξ ) are square matrices. The following isthe generalization of Turaev’s Theorem 2.2 to the noncommutative setting, and is stated in[Fri1, Theorem 2.1]. roposition 4.1 (cf [Tur, Theorem 2.2]. See also [Fri1, Theorem 2.1]) . Let ξ be a T -chainsuch that A i ( ξ ) is invertible for all odd i . Suppose that every B i is free. Then, A i ( ξ ) isinvertible for all even i if and only if H ∗ ( C ∗ ( X ; R )) = 0 . If H ∗ ( C ∗ ( X ; R )) = 0 , then T ( C i ( X ; R ) , {C i } ) = ε m Y i =0 A i ( ξ ) ( − i ∈ K ( R ) for some ε ∈ ± . Proof of Theorem 3.1.
We first recall the facts on invariance of Reidemeister torsion. Let Y be a finite connected finite CW complex. According to [Tur, Theorem 9.1], if there is a cellularmap f : Y → X which is homotopy equivalent, and H ∗ ( X ; R ) = 0 and the Whitehead groupof π ( Y ) is zero, then H ∗ ( Y ; R ) = 0 and the associated torsion T ( Y, ρ ′ ) is equal to T ( X, ρ ) in K ( R ) / ± ρ ( π ( X )) . Here, ρ ′ = ρ ◦ f ∗ . Furthermore, if X is homotopic to the knot complement S \ L , then the Whitehead group of π ( X ) vanishes by Waldhausen [Wal]. Therefore, it isenough for the proof to find X and Y whose homotopy type are S \ L such that T ( X, ρ ) · (1 − τ )equals Φ ρ,m and T ( Y, ρ )(1 − τ ) equals [ A F,W ]First, we take the CW-complex X corresponding with the Wirtinger presentation. Namely, X consists of a single vertex, m edges labeled by the generators x , . . . , x m and ( m −
1) 2-cellsattached by the relations r , . . . , r m − . As is known, we can easily verify that W is homotopicto S \ L . We regards these x , . . . , x m and r , . . . , r m − as cells of X . The cellular complex iswritten in0 → m − M j =1 A κ (( τ )) r j ∂ −→ m M i =1 A κ (( τ )) x i ∂ −→ A κ (( τ )) → , (6)as left A κ (( τ ))-modules where the boundary maps are given by ∂ ( r j ) = X i ρ (cid:16) ∂r j ∂x i (cid:17) x i , and ∂ ( x i ) = 1 − ρ ( x i ) . Notice that the restriction of ∂ on A κ (( τ )) x m is 1 − ρ ( x m ) = 1 − w − m τ w m ; hence, it is invertible.Therefore, the acyclicity of (6) is equivalent to the invertibility of Φ ρ,k . If we set ξ := { } and ξ := { , . . . , m } as in Proposition 4.1, the Reidemeister torsion T ( X, ρ ) is given byΦ ρ,m · (1 − τ ) − , as required.Next, we let Y be the CW complex corresponding with the presentation (1) in a similarway. Then, it is shown [Tro, Section 2.3] that this Y is known to be homeomorphic to S \ L ,and the chain complex is given by0 → g M j =1 A κ (( τ )) r Fj ∂ −→ g M i =1 A κ (( τ )) x i ⊕ A κ (( τ )) m ∂ −→ A κ (( τ )) → , where the boundary maps are given by ∂ ( r j ) = (1 − ρ ( y j )) m + X i ρ (cid:16) ∂r Fj ∂x i (cid:17) x i , and ∂ ( x i ) = 1 − ρ ( x i ) , ∂ ( m ) = 1 − τ. otice that the restriction of ∂ on A κ (( τ )) m is 1 − τ invertible. Therefore, similarly, Proposition4.1 implies that the Reidemeister torsion T ( Y, ρ ) is A F,W · (1 − τ ) − by definition, as required.This completes the proof. Proof of Proposition 2.4.
It is proven in the paper [Nos] that, K is fibered, if and only if thematrix A F,W is shown to be invertible for any ρ . Thus, Proposition 2.4 immediately deducesthe proof. Throughout this section, we assume the existence of m ∈ N such that κ m = id A . In this case,we suggest a relation to the torsions of the regular m -fold cyclic covering spaces E mL . Here the m -fold cyclic covering p : E mL → S \ L is associated with the surjection π ( S \ L ) → Z /m .For the purpose, we begin reviewing a ring homomorphism in [Nos]. For ℓ ≤ m and a ∈ A ,we let D ℓ ( a ) be a diagonal ( ℓ × ℓ )-matrix of the form D ℓ ( a ) := κ ℓ ( a ) 0 · · · κ ℓ − ( a ) · · · · · · κ ( a ) , and let O s,t be the zero ( s × t )-matrix. We define a square matrix M ℓ ( a ) := (cid:18) O m − ℓ,ℓ D m − ℓ ( κ ℓ +1 ( a )) D ℓ ( a ) O ℓ,m − ℓ (cid:19) ∈ Mat( m × m ; A ) . (7)Consider the Laurent polynomial ring A id (( t )), where t is a commutative indeterminate. Then,the paper [Nos] introduced a ring homomorphism Υ defined by settingΥ : A κ (( τ )) −→ Mat( m × m ; A id (( t ))); X j = k a j τ j X j = k M j ( a j ) t j . (8)Hence, we can consider the pushforwards of the K -classes: Υ ∗ ( T ( S \ L, ρ )) and Υ ∗ ( A F,W ).Here, by the Morita invariance on K I : K (Mat( m × m ; A id (( t )))) ∼ = K ( A id (( t ))) , the K -classes are regarded as quotient elements of K ( A id (( t ))) . Meanwhile, notice that the image of the composite ρ ◦ p ∗ : Z [ π ( E mL )] → A κ (( τ )) is containedin A κ (( τ m )) = A id (( t )). Thus, we can define the torsion T ( E mL , ρ ◦ p ∗ ) in K ( A id (( t ))). Theorem 5.1.
Let L be a knot. Under the above situation, the torsion of E mL is equal to thepushforward of the torsion of S \ L , that is, I ◦ Υ ∗ ( T ( S \ L, ρ )) = T ( E mL , ρ ◦ p ∗ ) ∈ K ( A id (( t ))) / ± ρ ◦ p ∗ ( π ( E mL )) . Proof.
We first analyze π ( E mL ). Recall from (1) the presentation π ( S \ L ) ∼ = h x , . . . , x g , m | r Fi := m − y i m z − i (1 ≤ i ≤ g ) i . or k ∈ Z /m , let x ( k ) i be a copy of x i , and y ( k ) i be the word obtained by replacing x i by x ( k ) i in the word y i . We similarly define the word z ( k ) i . Then, by a Reidemeister-Schreier method(see, e.g., [LS, Kab]), π ( E mL ) is presented by h x ( k )1 , . . . , x ( k )2 g , m ( k ∈ Z /m ) | r ( k ) i := m − y ( k ) i m ( z ( k +1) i ) − (1 ≤ i ≤ g, k ∈ Z /m ) i , and the injection p ∗ : π ( E mL ) → π ( S \ L ) is represented by the correspondence x ( k ) i x i , m m m . Then, similarly to (6), we have the cellular chain complex of C ∗ ( E mL ; A id (( t ))) as0 → m M j =1 2 g M k =1 A id (( t )) r ( k ) j ∂ −→ A id (( t )) m ⊕ m M i =1 2 g M k =1 A id (( t )) x ( k ) i ∂ −→ A id (( t )) → , where the boundary maps are given by ∂ ( r ( k ) j ) = (1 − ρ ( y ( k ) j )) m + X i,k ′ ρ (cid:16) ∂r ( k ) j ∂x ( k ′ ) i (cid:17) x ( k ′ ) i , and ∂ ( x ( k ) i ) = 1 − ρ ( x ( k ) i ) , ∂ ( m ) = 1 − t. Let J be the (2 gm × gm )-matrix { ρ (cid:0) ∂r ( k ) i /∂x ( k ′ ) j (cid:1) } i,j ≤ g, k,k ′ ≤ m . Then, similarly to the proofof Theorem 3.1, Proposition 4.1 implies T ( E mL , ρ ◦ p ∗ ) = (1 − t m ) − · J ∈ K ( A id (( t ))) / ± ρ ◦ p ( π ( E mL )) . (9)On the other hand, by the definition of Υ, we can easily checkΥ( ρ ( ∂r Fi ∂x j )) = n ρ ( ∂r ( s ) i ∂x ( t ) j ) o ≤ s,t ≤ m ∈ Mat( m × m ; A id (( t ))) , for any i, j ≤ m. Therefore, as (2 gm × gm )-matrices, Υ( A F,W ) = J , where A F,W is the matrixin (2). Notice that Υ(1 − τ ) = 1 − t m in K ( A id (( t ))), and recall T ( S \ L, ρ ) = (1 − τ ) − A F,W by the proof of Theorem 3.1. Hence, combing those with (9) deduces the required equality. K -Alexander invariants As is well-known, the (classical) Alexander polynomial of a knot has symmetry, i.e., it canbe expanded as P mi = − m a i t i such that a i ∈ Z and a − i = a i . Such a symmetry is called Reciprocity . As a generalization, if we choose an appropriate representation, the twistedAlexander polynomial also has reciprocity in some sense; see, e.g., [FKK, FV, Ki1, Kitan].Moreover, reciprocity of some Reidemeister torsions is generalized for K -groups; see [Mil1, §
10] or [Tur, Theorem 14.1]. Thus, it is reasonable to ask reciprocity on the K -classes ∆ K ρ .Unfortunately, some papers on reciprocity require some conditions to show duality theorems.Similarly, the author could not show directly reciprocity for every ∆ K ρ ; however, this paperwill observe (Theorem 6.1) a reciprocity of the pushforward by the ring homomorphism Υ ∗ ,under a certain situation. Here the point is to find a situation applicable to the theorem ofTuraev.The situation in this section is described as follows: ⋆ ) As in Example 2.1, we consider a group homomorphism h : π ( S \ L ) → H ⋊ Z suchthat H is a group of finite order, and h ( x i ) = ( g i ,
1) for some g i ∈ H . Let A be the group ring Q [ H ] over Q . Let m ∈ Z ≥ be the minimal such that κ m = id A .Starting from the situation ( ⋆ ), we discuss some involutions and observe some K . In thissection, let A be Q [ H ], and consider the (skew) Laurent polynomial ring A κ [ τ ± ], instead ofNovikov rings. Since A = Q [ H ] is semi-simple, the Wedderburn theorem immediately impliesthat there are division rings D , . . . , D m over Q and integers n , . . . , n k which ensure the ringisomorphism Q [ H ] ∼ = M i :1 ≤ i ≤ k Mat( n i × n i ; D i ) . For a division ring D , let D ( t ) be the fractional field of D [ t ± ] with ¯ t = t − and ι D : D [ t ± ] → D ( t ) be the inclusion. Then, the direct sum ⊕ ι D i gives rise to( ⊕ ι D i ) ∗ : Mat( m × m ; A id [ t ± ]) −→ M i :1 ≤ i ≤ k Mat( n i m × n i m ; D i ( t )) . Formally, we will denote by Q [ H ]( t ) by the direct sum ⊕ ki =1 D i ( t ). Then, Morita invariance on K -groups again implies the isomorphisms K ( M i :1 ≤ i ≤ k Mat( n i m × n i m ; D i ( t ))) ∼ = M i :1 ≤ i ≤ k K ( D i ( t )) ∼ = K ( Q [ H ]( t )) . By the assumption ( ⋆ ), A F,W in (2) is regarded as a matrix over the (uncompleted) Laurentpolynomial ring Q [ H ] κ [ τ ± ]. In the same way as (8), consider the ring homomorphism Υdefined by settingΥ : A κ [ τ ± ] −→ Mat( m × m ; A id ( t ± )); X i = k a i τ i X i = k M i ( a ) t i , where M j ( a j ) is defined in (14). Then, the pushforward ( ⊕ ι D i ) ∗ ◦ Υ ∗ ( A F,W ) can be regardedover Q [ H ]( t ). Thus, Υ ∗ ( A F,W ) can be considere to be an element of the K ( Q [ H ]( t )), wherewe omit writing ( ⊕ ι D i ) ∗ . Theorem 6.1.
Let l ∈ π ( S \ L ) be the preferred longitude of the knot L . Suppose the abovesituation ( ⋆ ), and invertibility of the pushforward Υ ∗ ( A F,W ) over Q [ H ]( t ) . In addition, weassume that either (a) Υ(1 − ρ ( l )) is an invertible matrix or (b) ρ ( l ) = 1 ∈ Q [ H ] .Then, under the isomorphism ( ⊕ ι D i ) ∗ , the following equality holds: Υ ∗ ( A F,W ) = Υ ∗ ( A F,W ) ∈ K ( Q [ H ]( t )) { K ( Q [ H ]) , t ℓ } ℓ ∈ Z . Remark 6.2.
As a result, we can observe the reciprocity on the K -class Υ ∗ (∆ K ρ ) in the K -group of a Novikov ring Q Q [ H ](( t )) , id as well. Indeed, since D i ( t ) and D i (( t )) are divisionrings, the associated inclusions j D i : D i ( t ) → D i (( t )) induce ι : Q [ H ]( t ) → Q [ H ](( t )); thus, thereciprocity inherits on ι ∗ (Υ ∗ ( A F,W )) in Q Q [ H ](( t )) , id .Furthermore, since the Dieudonn´e determinant over any division ring D induces an iso-morphism K ( D ) ∼ = ( D × ) ab , the K -group K ( Q [ H ]( t )) is identified with ( Q [ H ]( t ) × ) ab . Inparticular, if H is abelian, we can interpret Theorem 6.1 as the reciprocity of a polynomial. roof of Theorem 6.1. Let E mL be the cyclic covering space as above. For simplicity, let Υ ρ denote the composite Υ ◦ ρ . By virtue of Theorem 5.1, we may show only the reciprocity inReidemeister torsions of E mL .We first give the proof in the case (a). Then, [Tur, Corollary 14.2] immediately says T ( E mL , ∂E mL ; Υ ρ ) = T ( E mL ; Υ ρ ) ∈ K ( Q [ H ]( t )) / { K ( Q [ H ]) , τ ℓ } ℓ ∈ Z . (10)Moreover, thanks to the multiplicity of torsions with respect to short exact sequences (see[Tur, Theorem 3.4]), we immediately have T ( E mL , ∂E mL ; Υ ρ ) T ( ∂E mL ; Υ ρ ) = T ( E mL ; Υ ρ ) ∈ K ( Q [ H ]( t )) / { K ( Q [ H ]) , t ℓ } ℓ ∈ Z . (11)By Lemma 6.3 below, the second term T ( ∂E mL ; Υ ρ ) is equal to 1. Hence, the equalities (10)and (11) readily mean the required equality.Next, we discuss the case (b). Let M be the closed 3-manifold obtained by 0-surgery of E mL along L . Notice ∂M = ∅ . Let p : E mL → S \ L be the covering. Since ρ ( l ) = 1, thecomposite ρ ◦ p ∗ induces Z [ π ( M )] → A κ (( τ )) . Therefore, by [Tur, Corollary 14.2] again, wereadily have the reciprocity T ( M ; Υ ρ ◦ p ∗ ) = T ( M ; Υ ρ ◦ p ∗ ) . Hence, it is enough for the proof toshow (1 − t ) T ( M ; Υ ρ ◦ p ∗ ) = T ( E mL ; Υ ρ ) . To show this, consider the exact sequence appearing in a Mayer-Vietoris argument0 −→ C ∗ ( S × S ) ι ⊕ ι −→ C ∗ ( D × S ) ⊕ C ∗ ( E mL ) −→ C ∗ ( M ) −→ . Since 1 − Υ ρ ( l ) = 0, the boundary maps in the second C ∗ ( D × S ) are zero, and ι has asplitting as a chain map. Hence, the sequence is reduced to0 −→ C ∗ ( S ) ι −→ C ∗ ( E mL ) −→ C ∗ ( M ) −→ . Since the first and second complexes are acyclic, so is the third. Notice T ( C ∗ ( S )) = (1 − t ) bydefinition. Hence, by [Tur, Theorem 3.4] again, we obtain (1 − t ) T ( M ; Υ ρ ◦ p ∗ ) = T ( E mL ; Υ ρ )as required. Lemma 6.3. If − Υ ρ ( l ) ∈ Q [ H ] is invertible, then the torsion T ( ∂E mL ; Υ ρ ) is 1.Proof. Since ∂E mL is torus, we can describe the cellular complex as0 −→ Q [ H ]( t ) c − Υ ρ ( m ) , − Υ ρ ( l )) −−−−−−−−−−−→ Q [ H ]( t ) c ⊕ Q [ H ]( t ) c ′ Υ ρ ( l ) − − Υ ρ ( m ) ! −−−−−−−−→ Q [ H ]( t ) c −→ , which is acyclic by assumption. If we choose b = { c } , b = { c } , b = { c } , then [ b /c ] = 1 , [ b ∂ b /c ] = (cid:18) − Υ ρ ( l ) 11 − Υ ρ ( m ) 0 (cid:19) , [ b ∂ b /c ] = 1 − Υ ρ ( m ) . Hence, by the definition of T , we get the conclusion. Applications to m -fold metabelian Alexander polynomials. The paper [Nos] introduces an ( m -fold) metabelian Alexander polynomial. As applications ofthe previous sections, we will see properties of the polynomial.Let us recall the polynomial. Let L be a knot, and H be the torsion subgroup Tor( H ( e E mL ; Z )).Suppose that mH = H , e.g., the case m is a prime power. Then, the commutator subgroup of π ( S \ L ) surjects on H via an abelianization. Thus, we canonically have a group epimorphism ρ meta m : π ( S \ L ) → H ⋊ Z satisfying the assumption ( † ). Notice A = Q [ H ] is isomorphic to adirect sum of cyclotomic fields over Q ; see, e.g., [Tur, Corollary 12.10]. Then, the metabelianAlexander polynomial is defined to be the determinant of the pushforward Υ ∗ ( A F,W ):∆ ρ meta m := det(Υ ∗ ( A F,W )) ∈ Q [ H ]( t ) × / ( Q [ H ] × ) . As seen in Appendix A, this polynomial can be also interpreted as a torsion obtained from a S -valued Morse theory.As a slight generalization of Lemma 4 of [CG], we will show the invertibility of Υ ∗ ( A F,W ). Proposition 7.1 (cf. [CG]) . The matrix Υ ∗ ( A F,W ) in Mat(2 gm × gm ; Q [ H ]( t )) is invertible.Proof. Let S ⊂ H = Tor H ( E mL ; Z ) be any direct summand isomorphic to Z /p n for some n ∈ N and prime p . For the proof, we may replace H by S , since GL n ( Q [ H ⊕ H ]) = GL n ( Q [ H ]) × GL n ( Q [ H ]) for any finite abelian groups H i . Let e E L be the infinite cyclic covering space of S \ L . Let f M → e E L be the abelianfinite covering associated with the projection π ( e E L ) proj . −→ Tor H ( E mL ; Z ) proj . −→ S . Accordingto Lemma 4 and its corollary of [CG], the rational homology H ∗ ( f M ; Q ) is shown to be zero.Moreover, by Shapiro lemma (see, e.g., [KL]), the cellular complex C ∗ ( f M ; Q ) is isomorphicto C ∗ ( E mL ; Q [ S ][ t ± ]) of finite dimension. Thus, the tensored complex C ∗ ( E mL ; Q [ S ]( t )) = C ∗ ( E mL ; Q [ S ][ t ± ]) ⊗ Q [ S ]( t ) over Q [ S ]( t ) is acyclic. In usual, the first boundary ∂ is a splittablesurjection; hence, the acyclicity of C ∗ ( E mL ; Q [ S ]( t )) implies the invertibility of ∂ . As seen inthe proof of Theorem 3.1, ∂ is represented by the matrix Υ ∗ ( A F,W ). Hence, we complete theproof.As a corollary, we will see the reciprocity of ∆ ρ meta m . Here, notice that ρ ( l ) = 0 H since thepreferred longitude l is bounded by a Seifert surface in e E L . Therefore, our situation fulfillsthe conditions in Theorem 6.1; hence, we immediately have Corollary 7.2.
The metabelian Alexander polynomial has reciprocity in the sense of ∆ ρ meta m = ∆ ρ meta m ∈ Q [ H ]( t ) × / { Q [ H ] × , t ± } . This situation is similar to the Casson-Gordon invariant [CG], which discusses obstructionsof sliceness. Thus it is reasonable to consider applications for sliceness from our results.
Theorem 7.3 (cf. [KL, Theorem 6.2]) . Suppose that the knot L is topological slice, and H ( E mL ; Z ) is isomorphic to Z ⊕ T for some torsion module T .Then, there is a subgroup B ⊂ H = Tor H ( E mL ; Z ) such that | B | = | H | and that thereexists a polynomial f ( t ) ∈ Q [ H/B ]( t ) satisfying P ∗ (∆ ρ meta m ) = at n (1 − t ) f ( t ) f ( t ) ∈ Q [ H/B ]( t ) , (12) or some a ∈ Q [ H/B ] × , n ∈ Z , where P : H → H/B is the projection.
Remark 7.4.
We give a comparison with the works [KL, HKL]. The papers suppose m tobe a prime power, and consider a homomorphism χ : H ( B mL ; Z ) → Z /p d Z for some prime p .Choose a homomorphism λ : Z /p d Z → GL ( C ). Then, [KL, Theorem 6.2] claims that thefurther pushforward ( λ ◦ χ ) ∗ ◦ P ∗ (∆ ρ meta m ) is decomposed as (12). Thus, Theorem 7.3 is a slightgeneralization of [KL, Theorem 6.2], although the proof below is outlined on discussions in[CG, HKL, KL]. For the proof, we review Reidemeister torsions suitable to non-acyclic cases, in terms of de-terminants. Let F be a commutative field of characteristic zero. Let X be a finite connectedCW-complex X . Choose homomorphisms ρ : π ( X ) → F × and α : π ( X ) → Z = { t n } n ∈ Z . Wehave the tensor representation ρ ⊗ α : π ( X ) → F [ t ± ] × . Then, the cellular complex with localcoefficients is defined to be C ∗ ( X ; F ( t )) = F ( t ) ⊗ F [ Z ] C ∗ ( X ; F [ Z ]) = F ( t ) ⊗ ρ ⊗ α C ∗ ( e X ) . If C ∗ is a based chain complex, c i is a basis for C i , b i a basis for the boundaries B i , h i a basisfor the homology H i , then the Reidemeister torsion T ′ of the based chain complex is definedby T ( X, ρ, h ∗ ) ′ = Q i det[ b i e h i e b i /c i ] Q i det[ b i − e h i − e b i − /c i − ] ∈ F ( t ) × / { t n , a ∈ F × } . In this expression, e h i is a choice of lift of h i to C i , and e b i is a choice of b i to C i +1 using thedifferential ∂ i +1 : C i +1 → C i . It is known this T ( X, ρ, h ∗ ) ′ is independent of the choices of b i and of the lifts. Notice that, if C ∗ is acyclic and dim X ≤
3, then this T ( X, ρ, ∅ ) ′ is recoveredby the torsion in §
4: precisely, det T ( X, ρ ⊗ α ) = T ( X, ρ, ∅ ) ′ by definitions.In order to start the proof of Theorem 7.3, following [Mil1, Mil2], let us notice the followingtheorem (which is also stated in [KL, Theorem 5.1]). Theorem 7.5 (see [KL, Theorem 5.1]) . Let X be an oriented connected compact C ∞ -manifoldwith boundary ∂ ( X ) . Choose a triangulation of X as a CW-complex. Suppose that C ∗ ( X ; F ( t )) is not acyclic, but C ∗ ( ∂X ; F ( t )) is acyclic, and choose an appropriate basis h q for H q ( X ; F ( t )) for each q . Then, with respect to these bases, T ( ∂X, ρ ) ′ = T ( X, ρ, h ∗ ) ′ T ( X, ρ, h ∗ ) ′ ( − dim( X ) . Proof of Theorem 7.3.
We give some preparations. Let D ⊂ B be the slice disk of L in the 4-ball, and e B → B \ D be the m -fold covering. Let M (resp. X ) be the oriented closed manifoldobtained by 0-surgery of E mL along L (resp. of e B along D ). Then, we notice ∂X = M , andTor H ( M ; Z ) ∼ = H = Tor H ( e E mL ; Z ). Let B ⊂ H be the subgroup consisting of metabolizersof the linking form on M . Then, it is known (see [CG] or [KL, § | B | = | H | and theprojection ρ meta m : π ( M ) → H extends to π ( X ) → H/B . Recall that the group ring Q [ H/B ]is ring isomorphic to ⊕ i F i , where F i is a cyclotomic field over Q . Let q i : Q [ H/B ] → F i be theprojection. otice that we can find f i ∈ F i ( t ) satisfying q i ◦ T ( M , ρ ) ′ = a i t n i f i ( t ) f i ( t ) ∈ F i ( t ) , (13)for some a i ∈ F × i , n i ∈ Z . Indeed, we may only let ρ i be q i ◦ ρ meta m , and f i ( t ) be T ( X, ρ i , h ∗ ) inTheorem 7.5. Moreover, as in Theorem 6.1, each T ( M , ρ i ) ′ has reciprocity, we may suppose n = n = · · · . Furthermore, by the end of the proof of Theorem 6.1, we notice q i ◦P ∗ (∆ ρ meta m ) = T ( E mL , ρ i ) ′ = (1 − t ) T ( M , ρ i ) ′ . To conclude, since Q [ H/B ]( t ) ∼ = ⊕ i F i ( t ), the multiplicationsof (13) running over i implies the required equality (12). A Appendix; relation to S -valued Morse theoretic torsions Fix a preferred longitude l ∈ π ( S \ L ). Under an assumption ρ ( l ) = 1, the K -class Φ ρ,k canbe described from S -valued Morse theory; however, this section essentially contains nothingnew. In fact, this sections are analogous to [P, Ki2]. Here, we suppose notation in §§ S -valued Morse theory, and the main result of [P]; see[P] for the details. Let M be a connected closed C ∞ -manifold, and f : M → S be a Morsemap. Suppose that f ∗ : H ( M ) → H ( S ) = Z is surjective. As in Example 2.1, let G be asemidirect product H ⋊ Z with projection χ : G → Z , and A be Q [ H ]. Then, we have theNovikov ring A κ (( τ )).We study a logarithm from K ( A κ (( τ ))). For n ∈ Z , let Γ n be the set of conjugacy classescontained in χ − ( n ). Take the Q -vectors space Q Γ n spanned by Γ n , and define G by Q n ≥ Q Γ n .Furthermore, consider the multiplicative subgroup, W , of A κ (( τ )) × consisting of elements ofthe form 1 + a τ + a τ + · · · . The image of W in K ( A κ (( τ ))) will be denoted by c W . Thelogarithm log : W → G is defined bylog(1 + µτ ) = µτ − ( µτ ) · · · + ( − n − ( µτ ) n n + · · · with µ ∈ A κ [[ τ ]] . Then, it is shown [P, Lemma 1.1] that this map induces a homomorphism L : c W → G . Thesubgroup c W is known to be a direct summand of K ( A κ (( τ ))) [PR]; hence, L is regarded as ahomomorphism from K ( A κ (( τ ))).Next, we will review non-ablelian eta functions. For a vector field v , which satisfies a“Kupka-Smale condition”, let Cl( v ) be the closed orbits of v . For a closed orbit γ ∈ Cl( v ),let ǫ ( γ ) ∈ {± } denote the index of the corresponding Poincar´e map; let m ( γ ) denote themultiplicity of γ . Here is the definition of the non-ablelian eta function of − v : η L ( − v ) = X γ ∈ Cl( − v ) ǫ ( γ ) m ( γ ) { γ } ∈ G . Let denote G ( f ) be the set of vector fields satisfying the Kupka-Smale condition.The main theorem of [P] clarified the difference between torsions from the usual complexand from the Novikov complex. To be precise, there is an open dense subset G ( f ) ⊂ G ( f )with respect to a certain C -topology such that, for every v ∈ G ( f ), there is a chain homotopyequivalence φ : C ∗ ( v ) −→ C ∗ ( f M ; Z ) ⊗ Z [ H ] A κ (( τ )) uch that L ( τ ( φ )) = η L ( − v ) . Here τ ( φ ) is the torsion of the acyclic complex of the cokernel Coker( φ ).We will give a conclusion from the assumption ( ∗ ) and suppose ρ ( l ) = 1 as above (Forinstance, the setting in § M L, be the closed 3-manifold obtained by 0-surgery of S \ L along L . Then, ρ induces a homomorphism π ( M L, ) → A κ (( τ )). As in theproof of Theorem 6.1 implies the torsion T ( M L, , ρ ) is equal to (1 − t ) − T ( S \ L, ρ ). Then, bythe proof Theorem 3.1, we have [Φ ρ,k ] = (1 − t ) T ( M L, , ρ ) . To conclude, the torsions L ([Φ ρ,k ])can be described by the torsion defined from S -valued Morse complex and the eta function. B Relation to the higher order Alexander polynomial
The previous papers [C, Har] suggested a generalization of the classical Alexander polynomial;see also [GS, Fri2] for the studies. The point is that the generalized polynomial is defined froma group G , which is locally indicable and amenable, not any group. The purpose of this sectionis to show Proposition B.1 below, which suggests a relation between the generalized polynomialand the K -class ∆ K ρ . The facts and explanations in this section are essentially based on [Fri2];there is almost nothing new in this appendix. This appendix supposes terminology in §§
2– 4.For this, we start by reviewing the situation in [C, Har]. Let Ab : π ( S \ L ) → Z be anabelianization. Let G be a locally indicable and amenable group. It is known that the groupring Z [ G ] embeds in a (skew) fractional field. Fix an epimorphism φ : π ( S \ L ) → G suchthat there exists a group homomorphism ϕ G : G → Z satisfying Ab = ϕ G ◦ φ . Such a pair( ϕ G , φ ) is called an admissible pair for π ( S \ L ), following [Har, Definition 1.4].Let G ′ be the kernel Ker( ϕ G : G → Z ). Since G ′ is also locally indicable and amenable, Z [ G ′ ] embeds in a fractional field K G ′ . Let µ ∈ G be φ ( m ), and define γ : K G ′ → K G ′ tobe the homomorphism induced by γ ( g ) = µgµ − . Then, we obtain the skew polynomial ring K G ′ γ [ τ ± ], and a ring homomorphism ν : Z [ G ] −→ K G ′ γ [ τ ± ]; X g n g g X g n g gµ − φ ( g ) τ φ ( g ) . Let A be the division ring K G ′ , and κ be γ . Then, the following composite satisfies Assumption( † ): ρ G : Z [ π ( S \ L )] φ −→ Z [ G ] ν −→ K G ′ γ [ τ ± ] ֒ → K G ′ γ (( τ )) (14)Next, we review the higher order Alexander polynomial. The ring K G ′ γ [ τ ± ] is known tobe a principal ideal domain since K G ′ γ is a skew field. For a finitely generated right K G ′ γ [ τ ± ]-module H , the elementary divisor theorem claims an isomorphism H ∼ = M i : 1 ≤ i ≤ ℓ K G ′ γ [ τ ± ] /p i ( t ) K G ′ γ [ τ ± ]for some ℓ ∈ N and p i ( t ) ∈ K G ′ γ [ τ ± ] for i ∈ { , . . . , ℓ } . Following [C, Fri2], we define ord( H )by the product p ( t ) · · · p ℓ ( t ). Although it is a subject to discuss which set this ord( H ) shouldbe contained in, according to [Fri2], we regard ord( H ) as an element in K G ′ γ ( τ ) × ab ∪ { } up to ultiplication by an element of the form kτ e with k ∈ K G ′ γ and e ∈ Z . As is known, ord( H )depends on only H ; see [Fri2, Theorem 3.1]. Then, the order of the i -th homology∆ ψi := ord H i ( S \ L ; K G ′ γ [ τ ± ]) ∈ K G ′ γ ( τ ) × ab / { kτ e } k ∈ K G ′ γ ,e ∈ Z is called the higher order Alexander polynomial .To state Proposition B.1, let us consider the natural inclusion λ from the fractional field K G ′ γ ( τ ) into K G ′ γ (( τ )), since K G ′ γ (( τ )) is also a field. Then, by the isomorphism K ( K G ′ γ (( τ )) × ) ∼ =( K G ′ γ (( τ )) × ) ab , the inclusion yields the homomorphism λ ∗ : K G ′ γ ( τ ) × ab / { kτ e } k ∈ K G ′ γ ,e ∈ Z −→ Q A ,γ / { τ e } e ∈ Z . Then, the pushforward of ∆ ψ turns out to be almost the K -class ∆ K ρ G ; More precisely, Proposition B.1.
Suppose that the matrix A F,W is invertible. Then, ∆ ψi is zero if i > , and ∆ ψ = 1 − τ . Furthermore, the following equality holds: λ ∗ (∆ ψ ) = ∆ K ρ G ∈ Q A ,γ / { τ e } e ∈ Z . Proof.
From the cellular complex (6), we see that the i -th homology is vanishes for i >
1, i.e.,∆ ψi = 0, amd if i = 0, we find H ( S \ L ; K G ′ γ [ τ ± ]) ∼ = K G ′ γ [ τ ± ] / (1 − τ ) K G ′ γ [ τ ± ] . Thus, ∆ ψ = 1 − τ .Let us consider the case i = 1. Then, the main theorem of [Fri2], T ( S \ L, ρ G ) = ∆ ψ / ∆ ψ ∈ K G ′ γ ( τ ) × ab / { kτ e } k ∈ K G ′ γ ,e ∈ Z , where ρ G is the composite in (14). In the proof of Theorem 2.3, λ ∗ ( T ( S \ L, ρ G )) was shownto be [ A F,W ] · (1 − τ ) − . Hence, we have the required equality.However, the point is that, in general, the fractional field K G ′ is quite incomprehensiblefrom quantitative viewpoint; in particular, it is difficult to compute the abelianization K G ′ γ ( τ ) × ab and to distinguish whether two elements in K G ′ γ ( τ ) × ab are equal or not. By the reasons, it isbelieved that there is no example of computing their Alexander polynomial with non-trivialityeven if L is an easy knot.In contrast, in the definition of the K -class ∆ K ρ , we considered no fraction. In [Nos, §§ K ρ without using fraction, but using alogarithm and the homomorphism Υ ∗ . This is a reason why we employ Novikov rings insteadof fractional fields and polynomial ring. References [C] T. D. Cochran.
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