The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces

Murakami, Hitoshi; Yokota, Yoshiyuki

doi:10.1515/crelle.2007.045

Cited by 28 publications

(44 citation statements)

References 23 publications

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“…See, for example, [20] about recent developments of the conjecture and its generalizations. As one of the generalizations, Yokota and the author [22] proved that for the figure-eight knot the coloured Jones polynomial contains much more information. Actually, we showed that if we perturb the parameter 2π √ −1 a little, then the corresponding limit determines the SL(2; C) Chern-Simons invariant associated with an irreducible representation of π 1 (S 3 \ K) to SL(2; C) in the sense of Kirk and Klassen [15].…”

Section: Introductionmentioning

confidence: 99%

The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot

Murakami

2012

Journal of Topology

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Section: Introductionmentioning

confidence: 99%

The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot

Murakami

2012

Journal of Topology

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“…The original volume conjecture (Conjecture 0.1) for the figure-eight knot was prove by T. Ekholm (see for example [30]). Conjecture 0.2 was proved by Yokota and the author [35] in the case of the figure-eight knot. The following theorem appears in [33], proving Conjecture 0.3.…”

Section: Topological Interpretation Of the Asymptotic Expansion Of Thmentioning

confidence: 90%

“…The conjecture is true for the figure-eight knot [35]. This means that for large N we can write for a complex function S(u).…”

mentioning

confidence: 94%

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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“…It was demonstrated [8,22] that the A-polynomial has a relationship with an asymptotic limit of the colored Jones polynomial for a case of the figure-eight knot. Our purpose in this article is to show that this correspondence is also supported for a case of the twist knots and the torus knots.…”

Section: Imentioning

confidence: 99%

Asymptotics of the colored Jones polynomial and the A-polynomial

Hikami

2007

Nuclear Physics B

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ASYMPTOTICS OF THE COLORED JONES POLYNOMIAL AND THEA-POLYNOMIAL KAZUHIRO HIKAMI A. We reveal a relationship between the colored Jones polynomial and the A-polynomial for twist knots. We demonstrate that an asymptotics of the N-colored Jones polynomial in large N gives the potential function, and that the A-polynomial can be computed. We also discuss on a case of torus knots. IThe N-colored Jones polynomial J K (N) is a quantum invariant which is defined based on the N-dimensional irreducible representation of the quantum group U q (sl(2)). Motivated by Volume Conjecture raised by Kashaev [16], it was pointed out that the colored Jones polynomial at a specific value should be related to the hyperbolic volume of knot complement [21].As another example of the knot invariant related to S L(2; C), we have the A-polynomial [2,3]. This is defined as an algebraic curve of eigenvalues of the S L(2; C) representation of the boundary torus of knot, and contrary to the quantum invariants such as the colored Jones polynomial it includes many geometrical informations such as the boundary slopes of the knot.Those two knot invariants are superficially independent. Though, it is recently conjectured [4] that the homogeneous difference equation of the N-colored Jones polynomial for knot K with respect to N gives the A-polynomial for K (AJ conjecture). This fact was originally verified for both the trefoil and the figure-eight knot with a help of computer algebraic system [4], and was later proved for the torus knots [10]. We should note that in Ref. 26 a recursion relation of the summand of the colored Jones polynomial for the twist knots was shown to give the A-polynomial. See also Refs. 6, 7.Recently pointed out is still another connection between the colored Jones polynomial and the A-polynomial. It was demonstrated [8,22] that the A-polynomial has a relationship with an asymptotic limit of the colored Jones polynomial for a case of the figure-eight knot. Our purpose in this article is to show that this correspondence is also supported for a case of the twist knots and the torus knots.We recall a fact [5] that the N-colored Jones function J K (N) for knot K can be written in a form of the q-hypergeometric function. Once we obtain an invariant in the form of the q-hypergeometric series, we may define the H-function [20] for knot K based on the integrand of an asymptotics of

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The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces

Abstract: We calculate limits of the colored Jones polynomials of the figureeight knot and conclude that in most cases they determine the volumes and the Chern-Simons invariants of the three-manifolds obtained by Dehn surgeries along it.

Cited by 28 publications

References 23 publications

The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot

The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot

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

Asymptotics of the colored Jones polynomial and the A-polynomial

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