# Twisted Alexander invariants and hyperbolic volume.

1. Introduction. The purpose of this note is to give a formula of the hyperbolic volume of a knot complement using twisted Alexander invariants.A twisted Alexander polynomial was first defined in [3] for knots in the 3-sphere, and Wada ([10]) generalized this work and showed how to define a twisted Alexander polynomial given only a presentation of a group and representations to Z and GL(V) where V is a finite dimensional vector space over a field. In [2], Kitano proved that in the case of knot groups the twisted Alexander polynomial can be regarded as a Reidemeiser torsion.

Let M be a compact and oriented 3-manifold whose interior admits a finite volume hyperbolic structure. Porti ([8]) has investigated the Reidemeister torsion of M associated with the adjoint representation Ado[Hol.sub.M] of its holonomy representation [Hol.sub.M] : [[pi].sub.1](M) [right arrow] PSL(2, C), and then Yamaguchi showed in [13] a relationship between the Porti's Reidemeister torsion and the twisted Alexander invariant explicitly.

Muller's work ([7]) provides the relation between the Ray-Singer torsion and the hyperbolic volume of a compact hyperbolic 3-manifold. By another work ([6]) of Muller on the equivalence between the Reidemeister torsion and the RaySinger torsion for unimodular representations, we know the hyperbolic volume of a compact 3-manifold can be expressed using a Reidemeister torsion. After the works, Menal-Ferrer and Porti ([5]) obtained a formula of the volume of a cusped hyperbolic 3-manifold M using 'Higher-dimensional Reidemeister torsion invariants', which are associated with representations [[rho].sub.n] : [[pi].sub.i](M) [right arrow] SL(n, C) corresponding to the holonomy representation [Hol.sub.M] : [[pi].sub.1](M) [right arrow] PSL(2; C) (see Section 3 for the detail).

In this note, we show that the Yamaguchi's method in [12,13] is applicable to Higher-dimensional Reidemeister torsion invariants, so that we have a formula of the hyperbolic volume of a knot complement using twisted Alexander invariants. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (t) be the twisted Alexander invariant of Wada's notation ([10]). For the integer k(> 1), set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 1.1. Let K be a hyperbolic knot in the 3-sphere. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the last section, we give some calculations for the figure eight knot. The details, including link case, will be given elsewhere.

2. Reidemeister torsions and twisted Alexander invariants. Following [9] and [13], we review some definitions and conventions in this section.

Let F be a field and [C.sub.*] = ([C.sub.*], [partial derivative]) a chain complex of finite dimensional F-vector spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For each i, we denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the homology is denoted by [H.sub.i] = [Z.sub.i]/[B.sub.i]. By the definition of [Z.sub.i], [B.sub.i] and [H.sub.i], we obtain the following exact sequence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [[??].sub.i-1] be a lift of [B.sub.i-1] to [C.sub.i], and [[??].sub.i] a lift of [[??].sub.i] to [Z.sub.i]. Then we can decompose [C.sub.i] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [c.sup.i] be a basis for [C.sub.i] and c the collection [{[c.sup.i]}.sub.i[greater than or equal to] 0]. Similarly, let [h.sup.i] be a basis for [H.sub.i], if nonzero, and h the collection [{[h.sup.i]}.sub.i[greater than or equal to] 0]. We choose [b.sup.i] a basis of [B.sub.i]. Let [[??].sup.i-1] be a lift of [b.sup.i-1] to [C.sub.i], and [[??].sup.i] lift of [h.sup.i] to [Z.sub.i], then we have a new basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] means a disjoint union. We denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the determinant of the transformation matrix from the basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.1. The torsion of the chain complex [C.sub.*] with basis c and h for [H.sub.i] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is known that tor([C.sub.*], c, h) is independent of the choice of bi and the lifts [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 2.2. In [5], Menal-Ferrer and Porti use [(-1).sup.i] instead of [(-1).sup.i+1] in Definition 2.1. Then the sign of the right-hand side of the equation in Theorem 7.1 in [5] becomes opposite. See Remark 2.2 and Theorem 4.5 in [9].

Let W be a finite CW-complex, and [rho] : [[pi].sub.1](W, *) [right arrow] SL(n, F) a representation of its fundamental group. Consider the chain complex of vector spaces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.*]([??], Z) denotes the simplicial complex of the universal covering of W and [[cross product].sub.[rho]] means that one takes the quotient of [F.sup.n] [[cross product].sub.Z] [C.sub.*]([??]; Z) by Z-module generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The boundary operator is defined by linearity and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We denote by [H.sup.*](W, [rho]) the homology of this complex.

Let {[v.sub.1], ..., [v.sub.n]} be a basis of [F.sup.n] and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. denote the set of i-dimensional cells of W. We take a lift [[??].sup.i.sub.j] of the cell [c.sup.i.sub.j] in [??]. Then, for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a basis of the Z[[[pi].sub.1](W)]-module [C.sub.i]([??]; Z). Thus we have the following basis of [C.sub.i](W, [rho]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose [H.sub.1](W,[rho]) = 0, and let [h.sup.1] be a basis of [H.sub.1](W; [rho]). We denote by h the basis {[h.sup.0], ..., [h.sup.dimW]} of [H.sub.1](W,[rho]). Then tor([C.sub.*](W,[rho]), c, h)([member of] [F.sup.*]/{[+ or -]1}) is well defined. Note that it does not depend on the lifts of the cells [[??].sup.i] since det [rho] = 1. Further, if the Euler characteristic of W is equal to zero (e.g. the case that W corresponds to a knot exterior), it does not depend on the choice of a basis {[v.sub.1], ..., [v.sub.n]} (cf. Lemma 2.4.2 in [13]).

Remark 2.3. The Reidemeister torsion is independent of the choice of a base point * of the fundamental group [[pi].sub.1](W, *). Furthermore, it is known that the Reidemeister torsion is an invariant under subdivision of the cell decomposition of W with p-coefficients up to factor [+ or -]1.

Remark 2.4. Let K be a knot in the 3-sphere [S.sup.3] and [M.sub.K] = [S.sup.3]--int N(K). We denote by G(K) the fundamental group of [M.sub.K]. From the result of Waldhausen ([11]), the Whitehead group Wh(G(K)) is trivial. In such a case, the Reidemeister torsion does not depend on the choice of its CW-structure. Suppose [H.sub.1]([M.sub.K,[rho]]) = 0. Then the Reidemeister torsion does not depend on h = 0. In this case we denote by tor([M.sub.K, [rho]]) the Reidemeister torsion.

Let [alpha] be a surjective homomorphism from [[pi].sub.1] (W, *) to the multiplicative group (t). Instead of a representation [rho] : [[pi].sub.1](W, *) [right arrow] SL(n, F), consider the twisted representation:

[alpha] [cross product] [rho] : [[pi].sub.1](W, *) [right arrow] GL(F(t)),

where F(t) is the field of fraction of the polynomial ring F[t]. By the same method as above, we can define tor([C.sub.*](W, [alpha] [cross product] [rho]), 1 [cross product] c, h) ([member of] [F.sup.*](t)/{[+ or -][t.sup.nZ]}). As the determinant is not one, there is an independency factor [t.sup.nm], for some integer m. More precisely, we define:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the action is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The boundary operator is defined by linearity and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Kitano ([2]) investigated the relationship between the Reidemeister torsions and the twisted Alexander invariants for knots. Namely, he proved

Theorem 2.5 ([2]). Let K be a knot in the 3-sphere [S.sup.3] and [m.sub.K] = [S.sup.3]--intN(K). Suppose [rho] is a non-trivial representation such that [H.sub.1]([M.sub.K],[rho]p) = 0. Then, [H.sub.1]([M.sub.K], [alpha] [cross product] [rho]) = 0 and tor([M.sub.K], [alpha] [cross product] [rho]) = [[DELTA].sub.K,[rho]](t), where [[DELTA].sub.K,[rho]](t) is the twisted Alexander invariant.

See also Theorem 2.13 in [9]. The twisted Alexander invariant can be computed using the Fox calculus ([1,2,10]).

3. Representations of the fundamental groups of hyperbolic 3-manifolds. Let M be an oriented, complete, hyperbolic 3-manifold of finite volume. Then M has the holonomy representation: [Hol.sub.M] : [[pi].sub.1](M, *) [right arrow] [Isom.sup.+][H.sup.3], where [Isom.sup.+][H.sup.3] is the orientation preserving isometry group of hyperbolic 3-space [H.sup.3]. Using the upper half-space model, [Isom.sup.+] [H.sup.3] is identified with PSL(2, C) = SL(2, C)/{[+ or -]1}. It is known that [Hol.sub.M] can be lifted to SL(2, C), and such lifts are in canonical one-to-one correspondence with spin structures on M. Thus, attached to a fixed spin structure [eta] on M, we get a representation:

[Hol.sub.(M,[eta])] : [[pi].sub.1]((M,[eta]), *) [right arrow] SL(2, C).

Let W be a finite CW-complex and [rho] a representation of [[pi].sub.1](W, *) to SL(2, C). Then the pair ([C.sup.2], [rho]) is an SL(2, C)-representation of [[pi].sub.1](W, *) by the standard action SL(2, C) to [C.sup.2]. It is known that the pair of the symmetric product [Sym.sup.n-1]([C.sup.2]) and the induced action by SL(2, C) gives an n-dimensional irreducible representation of SL(2, C). More precisely, let [V.sub.n] be the vector space of homogeneous polynomials on [C.sup.2] with degree n - 1, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the symmetric product [Sym.sup.n-1]([C.sup.2]) can be identified with [V.sub.n] and the action of A [member of] SL(2, C) is expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a homogeneous polynomial and the right-hand side is determined by the action of [A.sup.-1] on the column vector as a matrix multiplication. We denote by ([V.sub.n], [[sigma].sub.n]) the representation given by this action of SL(2, C) where [[sigma].sub.n] means the homomorphism from SL(2, C) to GL([V.sub.n]). It is known that each representation ([V.sub.n], [[sigma].sub.n]) turns into an irreducible SL(n, C)-representation of SL(2, C) and that every irreducible n-dimensional representation of SL(2, C) is equivalent to ([V.sub.n], [[sigma].sub.n]). Composing [Hol.sub.(M,[eta])] with [[sigma].sub.n], we obtain the following representation:

[[rho].sub.n] : [[pi].sub.1]((M,[eta]), *) [right arrow] SL(n, C).

In the following sections, we will discuss Reidemeister torsions associated with this representation [[rho].sub.n]. Note that there are several computations of the Reidemeister torsions associated with [[sigma].sub.2k] in [14,15].

4. The results of Menal-Ferrer and Porti. In this note, we focus on a knot complement. We introduce the results of Menal-Ferrer and Porti ([4,5]) in this setting.

Let K be a hyperbolic knot in the 3-sphere [S.sup.3], that is, [S.sup.3]-K is an oriented, complete, finite-volume hyperbolic manifold with only one cusp. Then, [S.sup.3]--K may be regarded as the interior of a compact manifold [M.sub.K] such that [partial derivative][M.sub.K] = T where T is homeomorphic to a torus [T.sup.2]. In what follows, we consider the compact manifold [M.sub.K] instead of [S.sup.3]--K.

By Corollary 3.7 in [4], we have that dimC [H.sup.i]([M.sub.K],[[rho].sub.2]) = 0 (i = 0,1, 2) if n is even, and that dimC [H.sup.0]([M.sub.K], [[rho].sub.2])= 0, dimC [H.sup.1]([M.sub.K], [[rho].sub.2]) = dimC [H.sup.2]([M.sub.K], [[rho].sub.2]) = 1 if n is odd. Further, in [5], Menal-Ferrer and Porti proved the following. (Note that Poincare duality with coefficients in [[rho].sub.2] holds (Corollary 3.7 in [5]).)

Proposition 4.1 (Proposition 4.6 in [5]). Suppose that [H.sub.*](T; [[rho].sub.2]) [not equal to] 0. Let G < [[pi].sub.1]([M.sub.K], *) be some fixed realization of the fundamental group of T as a subgroup of [[pi].sub.1]([M.sub.K], *). Choose a non-trivial cycle [theta] [member of] [H.sub.1](T; Z), and a non-trivial vector v [member of] [V.sub.n] fixed by [[rho].sub.2](G). Then the following holds: (a) A basis for [H.sub.1] ([M.sub.K], [[rho].sub.2]) is given by [i.sub.*]([v [cross product] [??]]).

(b) A basis for [H.sup.2]([M.sub.K], [[rho].sub.2]) is given by [i.sub.*]([v [cross product] [??]]). Here, i : T [??] [M.sub.K] denotes the inclusion.

Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. On the other hand, Menal-Ferrer and Porti (Theorem 0.2 in [4]) proved that [H.sub.1]([M.sub.K],[[rho].sub.2k]) = 0 for k [greater than or equal to] 1. Therefore, we may define the following quotients:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The quantity [T.sup.2k+1] is independent of the spin structure because of the fact that an odd-dimensional irreducible complex representation of SL(2, C) factors through PSL(2, C). Since [S.sup.3] - K has only one cusp, then all spin structures on [M.sub.K] are acyclic (Corollary 3.4 in [5]). This means that the limit of [T.sup.2k] is also independent of the spin structure (Theorem 7.1 in [5]). Thus it is not necessary to consider a spin structure on [M.sub.K] in our setting. Hence, the above definition may be simplified to the following form deleting

Definition 4.2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that it is proved that the quotient is independent of the choices h (Proposition 4.2 in [5]). Then, we can reduce Theorem 7.1 in [5] to the following statement:

Theorem 4.3 (Theorem 7.1 in [5]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As in Remark 2.2, the sign of the right-hand side is plus.

5. Proof of Theorem 1.1.

Case 1. Even-dimensional representation [[rho].sub.2k] case.

By Theorem 0.2 in [4], [H.sup.*]([M.sub.K], [[rho].sub.2k]) = 0 for k [greater than or equal to] 1. Then, by Theorem 2.5, we can prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1) from the map at the chain level [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] induced by evaluation t = 1. Then, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence we have done in the case of [[rho].sub.2k] in Theorem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by Theorem 4.3.

Case 2. Odd-dimensional representation [[rho].sub.2k+1] case.

Although the idea of the proof is the same as Yamaguchi's one in [12,13], I think it is worth outlining it here for the convenience of readers. He investigated the case of the adjoint representation of SL(2; C), which is essentially equivalent to [[rho].sub.3] in our setting.

The homology group [H.sub.*]([M.sub.K]; Z) = [H.sup.0]([M.sub.K]; Z) [[direct sum] [H.sub.1]([M.sub.K]; Z) has the basis {[p], [[mu]]}, where [p] is the homology class of a point and [[mu]] is that of the meridian of K. Further, [H.sub.1]([partial derivative][M.sub.K]; Z) has the basis {[[mu]], [[lambda]]}, where [[lambda]] is the homology class of a longitude of K. By Proposition 4.1, we may define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is known that [M.sub.K] collapses to a 2-dimensional CW-complex W with only one vertex. We call [phi] this deformation. Thus [M.sub.K] is simple homotopy equivalent to W. It is enough to prove the theorem for W since a Reidemeister torsion is a simple homotopy invariant.

By Proposition 3.5 in [1], we have [H.sup.0](W, [alpha] [cross product] [[rho].sub.2k+1])= 0. Further, we have the next lemma by the same argument as Proposition 7 in [12] or Proposition 3.1.1 in [13].

Lemma 5.1. For * = 1,2, we have: [H.sub.*]([M.sub.K], [alpha] [cross product] [[rho].sub.2k+1]) = 0.

Proposition 5.2. tor([M.sub.K], [alpha] [cross product] [[rho].sub.2k+1]) has a simple zero at t = 1. Moreover the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. We define the subchain complex [C'.sub.*](W, [[rho].sub.2k+1]) of the chain complex [C.sub.*](W,[[rho].sub.2k+1]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [C'.sub.*](W,[[rho].sub.2k+1]) = 0 (I [not equal to] 1,2). Note that v is fixed by [[rho].sub.2k+1] (G), and the boundary operators of [C'.sub.*] (W, [[rho].sub.2k+1]) are zero by the definition. The modules of this subchain complex are lifts of homology groups [H.sub.*](W, [[rho].sub.2k+1]). Similarly, we define the subcomplex [C.sub.*](W, [alpha] [cross product] [[rho].sub.2k+1]) of [C.sub.*](W, [alpha] [cross product] [[rho].sub.2k+1]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [C'.sub.i](W, [alpha] [cross product] [[rho].sub.2k+1]) = 0 for i [not equal to] 1, 2. Since v is an invariant vector of [[rho].sub.2k+1](G), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus the boundary operators of [C'.sub.*](W, [alpha] [cross product] [[rho].sub.2k+1]) are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This means that the homology of [C'.sub.*](W, [alpha] [cross product] [[rho].sub.2k+1]) is zero.

By the definition, the chain complex [C'.sub.*](W, [[rho].sub.2k+1]) has the natural basis:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [C".sub.*](W,[[rho].sub.2k+1]) be the quotient of [C.sub.*](W,[[rho].sub.2k+1]) by [C'.sub.*](W, [[rho].sub.2k+1]), c" a basis of [C".sub.*](W, [[rho].sub.2k+1]), and [[bar.c]" a lift of c" to [C.sub.*](W,[[rho].sub.2k+1]). By Lemma 5.1, we can apply Proposition 3.3.1 in [13] to this setting, then we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the calculation above, we have tor([C'.sub.*](W, [alpha] [cross product] [[rho].sub.2k+1]), 1 [cross product] c') = t - 1, thus we have this proposition.

Proof of Theorem 1.1. By Theorem 2.5 and Lemma 5.1, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We also have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by Proposition 5.2, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (t) is a rational function. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we have Theorem 1.1 by Theorem 4.3.

6. Some calculations on the figure eight knot complement. Let K be the figure eight knot [4.sub.1]. Note that it is known that the volume of K is 2.02988 .... The knot group G(K) has the following presentation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where a and b correspond to the meridians of K. Consider the representation of this fundamental group:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where u is a complex value satisfying [u.sup.2] + u + 1 = 0. This representation is the holonomy representation of G(K). By the definition, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the same calculations, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Via Fox calculus for G(K), we obtain the denominator of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. On the other hand, the numerator of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1). Here we use the value u = -1 + [square root of -3]. Continuing in this way, we have obtained the following data:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

n 4[pi] log [absolute n 4[pi] log [absolute value of [A.sub.K,n] value of [A.sub.K,n] (1)]/[n.sup.2] (1)]/[n.sup.2] 6 1.35850 ... 7 1.58331 ... 8 1.66441 ... 9 1.76436 ... 10 1.79618 ... 11 1.85105 ... 12 1.86678 ... 13 1.90158 ... 14 1.91009 ... 15 1.93361 ...

These calculations were done by using Wolfram Mathematica.

Acknowledgements. The author wishes to express his thanks to Prof. Yoshikazu Yamaguchi for many helpful conversations. He would like to thank an anonymous referee for his/her appropriate advice. He also thanks Profs. Takahiro Kitayama, Takayuki Morifuji and Joan Porti for several comments.

This work was supported by JSPS KAKENHI Grant Numbers JP15K04868.

References

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[8] J. Porti, Torsion de Reidemeister pour les variates hyperboliques, Mem. Amer. Math. Soc. 128 (1997), no. 612, x+139 pp.

[9] J. Porti, Reidemeister torsion, hyperbolic three-manifolds, and character varieties, arXiv: 1511.00400.

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[12] Y. Yamaguchi, On the non-acyclic Reidemeister torsion for knots, 2007. (Dissertation at the University of Tokyo).

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By Hiroshi GODA

Department of Mathematics, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan

(Communicated by Kenji Fukaya, M.J.A., June 13, 2017)

2010 Mathematics Subject Classification. Primary 57M27, 57M25, 57M50.

doi: 10.3792/pjaa.93.61

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Author: | Goda, Hiroshi |
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Publication: | Japan Academy Proceedings Series A: Mathematical Sciences |

Date: | Jul 1, 2017 |

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