Optimistic limits of the colored Jones polynomials
type: Item
instance of: scholarly article
zbMATH Open document ID: 1278.57018
DOI: 10.4134/JKMS.2013.50.3.641
author: Jinseok Cho
author: Jun Murakami
published in: Journal of the Korean Mathematical Society
publication date: 2013-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.3137
full work available at URL: http://mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 51M25
Mathematics Subject Classification ID: 58J28
zbMATH DE Number: 6196800
zbMATH Keywords: volume conjecture
zbMATH Keywords: colored Jones polynomial
zbMATH Keywords: hyperbolic volume
zbMATH Keywords: Kashaev invariant
zbMATH Keywords: Chern-Simons invariant
MaRDI profile type: Publication
arXiv ID: 1009.3137
OpenAlex ID: W2963516494
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title: Optimistic limits of the colored Jones polynomials
review text: For a link \(L\) in the three-sphere \(S^3\) and an integer \(N\geq2\), let \(\langle L\rangle_N\) be \textit{R. M. Kashaev}'s invariant [Mod. Phys. Lett. A 10, No. 19, 1409--1418 (1995; Zbl 1022.81574)]. Kashaev conjectured that \(2\pi\lim_{N\to\infty}\log|\langle K\rangle_N|/N\) would equal the hyperbolic volume of \(S^3\setminus{K}\) when \(K\) is a hyperbolic knot [\textit{R. M. Kashaev}, Lett. Math. Phys. 39, No. 3, 269--275 (1997; Zbl 0876.57007)] by giving some evidence, which is due to non-rigorous calculation, for hyperbolic knots with small crossing numbers. It was proved by J.~Murakami and the reviewer that Kashaev's invariant is the colored Jones polynomial \(J_N(L;q)\) associated with the \(N\)-dimensional irreducible representation of \(sl_2(\mathbb{C})\) evaluated at the \(N\)-th root of unity, who also proposed the volume conjecture for any knot replacing the hyperbolic volume in Kashaev's conjecture with the simplicial volume [\textit{H. Murakami} and \textit{J. Murakami}, Acta Math. 186, No. 1, 85--104 (2001; Zbl 0983.57009)].NEWLINENEWLINED.~Thurston observed that by considering a triangulation of a knot complement by ideal tetrahedra, five for each vertex of a knot diagram, one could prove the volume conjecture [\textit{D. Thurston}, ``Hyperbolic volume and the Jones polynomial'', Lecture notes, École d'été de Mathématiques `Invariants de nœuds et de variétés de dimension \(3\)', Institut Fourier - UMR 5582 du CNRS et de l'UJF Grenoble (France) du 21 juin au 9 juillet 1999, {\texttt{http://www.math.columbia.edu/\~{}dpt/speaking/Grenoble.pdf}}]. Each tetrahedron corresponds to a quantum factorial in the \(R\)-matrix defining the colored Jones polynomial. Y.~Yokota considered a similar triangulation using four tetrahedra around a vertex, each tetrahedron corresponding to a quantum factorial in Kashaev's \(R\)-matrix [\textit{Y. Yokota}, J. Knot Theory Ramifications 20, No. 7, 955--976 (2011; Zbl 1226.57025)].NEWLINENEWLINEIn both methods, replacing…
reviewed by: Hitoshi Murakami
description: scientific article; zbMATH DE number 6196800