Non Abelian Twisted Reidemeister Torsion for Fibered Knots

Dubois, Jérôme

doi:10.4153/cmb-2006-006-0

Cited by 29 publications

(48 citation statements)

References 16 publications

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“…Suppose that we are given a compact Lie group G with Lie algebra g. Let K be a knot in the three-sphere S 3 and V an irreducible representation of G. Let A be the set of all G-connection on the trivial G-bundle over where W V (K; A) is the Wilson loop, that is, the trace of the image in V by the representation of the element in G given by the parallel transport along K using the connection A. If G = SU (2) and V is the N -dimensional irreducible representation, this defines the N -dimensional colored Jones polynomial J N K; exp(2π √ −1/(k + 2)) .…”

Section: Introductionmentioning

confidence: 99%

“…(We need a cohomological orientation to define the sign but in this paper we do not need it. See [34] and [3] for details. )…”

Section: Introductionmentioning

confidence: 99%

“…It is known that for a torus knot, any irreducible representation is both µ-regular and λ-regular, where µ is the meridian, a loop that goes around the knot so that its linking number with the knot is one, and λ is the preferred longitude, a loop that goes along the knot so that its linking number with the knot is zero [3,Example 1]. It is also known that for a hyperbolic knot K, then an irreducible representation that defines a hyperbolic Dehn surgery is γ-regular, where γ is the simple closed curve on ∂E K along which the surgery is performed [30].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Representations and the colored Jones polynomial of a torus knot

Andersen¹,

Bodén²,

Hahn³

et al. 2011

Chern-Simons Gauge Theory: 20 Years After

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Section: Introductionmentioning

confidence: 99%

“…(We need a cohomological orientation to define the sign but in this paper we do not need it. See [34] and [3] for details. )…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Representations and the colored Jones polynomial of a torus knot

Andersen¹,

Bodén²,

Hahn³

et al. 2011

Chern-Simons Gauge Theory: 20 Years After

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show abstract

“…Here we assume that our representation is both λ-regular and µ-regular. It is known that any irreducible representation of π 1 S 3 \ T (a, b) is λ-regular and µ-regular (see [3,Example 1]). So the representation ρ u,ω1 given in Subsection 1.2 is λ-regular and µ-regular unless…”

Section: 5mentioning

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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“…Other explicit computations of the volume form ! K are given in [5] for fibered knots-and in particular for all torus knots-using the non abelian Reidemeister torsion.…”

Section: An Explicit Computationmentioning

confidence: 99%