Twisted Alexander polynomial and Reidemeister torsion

Kitano, Teruaki

doi:10.2140/pjm.1996.174.431

Cited by 106 publications

(110 citation statements)

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“…More precisely, we show that the Reidemeister torsion of a fibered knot defined for a certain tensor representation is expressed as a rational function of monic polynomials. This Reidemeister torsion is nothing but Wada's twisted Alexander polynomial (see [7] for details), so that our result can be regarded as a natural generalization of the property on the classical Alexander polynomial mentioned above. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 74%

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Reidemeister torsion, twisted Alexander polynomial and fibered knots

Okuno

Kitano

Morifuji

2005

Comment. Math. Helv.

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Section: Introductionmentioning

confidence: 74%

“…The first two assertions are nothing but [7], Proposition 3.1. The independence on j follows from [7], Lemma 1.2. Next, if we consider well-definedness up to ±t nk (k ∈ Z), we only have to recall Remark 2.4.…”

Section: Remark 24 It Is Known That τ ρ (X)mentioning

confidence: 94%

Reidemeister torsion, twisted Alexander polynomial and fibered knots

Okuno

Kitano

Morifuji

2005

Comment. Math. Helv.

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“…Theorem 3.7 ( [22,21]). Let K be a knot and ρ : π 1 (S 3 \K) → SL(2; C) a λ-regular representation, where λ is the preferred longitude.…”

Section: Theorem 34 ([46 Theorem 312])mentioning

confidence: 99%

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

Murakami

2014

Acta Math Vietnam

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Abstract. A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern-Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums and each summand is related to the Chern-Simons invariant and the Reidemeister torsion associated with a representation.Let J N (K; q) ∈ Z[q, q −1 ] be the colored Jones polynomial of a knot K in the three-sphere S 3 associated with the irreducible N -dimensional representation of the Lie algebra sl 2 (C) [14,19]. We normalize it so that J N (unknot; q) = 1. Note that J 2 (K; q) is the original Jones polynomial. R. Kashaev conjectured [16] that his knot invariant K N ∈ C introduced in [15] would grow exponentially with growth rate the volume of the knot complement S 3 \ K when the integer parameter N goes to the infinity if the knot K is hyperbolic. J. Murakami and the author [34] proved that Kashaev's invariant coincides with J N K; exp(2π √ −1/N ) and generalized Kashaev's conjecture to general knots.

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“…Let K be the knot 7 3 in Rolfsen's table, which is the 2-bridge knot K(13, 9). Its Alexander polynomial is ∆ K (t) = 2 − 3t + 3t 2 − 3t 3 + 2t 4 and thus the genus of K is two. We fix a presentation of the knot group G(K):…”

Section: Examplesmentioning

confidence: 99%