Optimistic limits of the colored Jones polynomials and the complex volumes of hyperbolic links
type: Item
instance of: scholarly article
zbMATH Open document ID: 1346.57020
DOI: 10.1017/S144678871600001X
author: Jinseok Cho
published in: Journal of the Australian Mathematical Society
publication date: 2016-06-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.3701
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 51M25
Mathematics Subject Classification ID: 58J28
zbMATH DE Number: 6591363
zbMATH Keywords: volume conjecture
zbMATH Keywords: optimistic limit
zbMATH Keywords: colored Jones polynominal
zbMATH Keywords: complex volume
MaRDI profile type: Publication
arXiv ID: 1303.3701
cites work: THE COMPLEX VOLUMES OF TWIST KNOTS VIA COLORED JONES POLYNOMIALS
cites work: Optimistic limits of the colored Jones polynomials
cites work: Optimistic limit of the colored Jones polynomial and the existence of a solution
cites work: The volume and Chern-Simons invariant of a representation
cites work: ON THE COMPLEX VOLUME OF HYPERBOLIC KNOTS
cites work: Pseudo-developing maps for ideal triangulations I: Essential edges and generalised hyperbolic gluing equations
cites work: THE COLORED JONES POLYNOMIALS OF 2-BRIDGE LINK AND HYPERBOLICITY EQUATIONS OF IT'S COMPLEMENTS
cites work: Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links
cites work: The colored Jones polynomials and the simplicial volume of a knot
cites work: The evolution of geometric structures on 3-manifolds
cites work: The hyperbolic volume of knots from the quantum dilogarithm
cites work: Braids, complex volume and cluster algebras
cites work: Generalized volume conjecture and the \(A\)-polynomials: The Neumann-Zagier potential function as a classical limit of the partition function
cites work: The complex volumes of twist knots
OpenAlex ID: W2963775364
Recommended article: Optimistic limits of the colored Jones polynomials
Recommended article: A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial
Recommended article: On the limit of the colored Jones polynomial of a non-simple link
Recommended article: Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links
Recommended article: Optimistic limit of the colored Jones polynomial and the existence of a solution
Recommended article: The colored Jones polynomials and the simplicial volume of a knot
Recommended article: Coloured Jones invariants of links and the Volume Conjecture
Recommended article: THE COMPLEX VOLUMES OF TWIST KNOTS VIA COLORED JONES POLYNOMIALS
Recommended article: Asymptotics of the colored Jones polynomial and the A-polynomial
Recommended article: Asymptotics of the colored Jones function of a knot
title: Optimistic limits of the colored Jones polynomials and the complex volumes of hyperbolic links
review text: The volume conjecture predicts that the volume \(\mathrm{vol}(L)\) of a hyperbolic link \(L\) should agree with the limit \(\lim_{N\to\infty}\frac{2\pi\log \mid \langle L\rangle_N \mid}{N}\) of the Kashaev invariant. The paper under review provides a more combinatorial approach to the ``optimistic limit'' of the Kashaev invariant that the author and and \textit{J. Murakami} introduced in [J. Korean Math. Soc. 50, No. 3, 641--693 (2013; Zbl 1278.57018)], modifying an earlier approach of \textit{Y. Yokota} [On the volume conjecture for hyperbolic knots, {\url arXiv:math/0009165}].NEWLINENEWLINEThe approach uses a link diagram and the associated octahedral triangulation of \(S^3\setminus(L\cup\left\{\pm\infty\right\})\). To each region of the link diagram one associates a number \(w_i\), and to each crossing of the link diagram one associates a ``potential'' which is a certain combination of dilogarithmic functions of combinations of the values of \(w_i\) for the adjacent regions. The potential \(W\) is then defined as the sum of the potentials over all crossings, and the author considers a modified potential by \(W_0(w_1,\ldots,w_n)=W(w_1,\ldots,w_n)-\sum_{k=1}^nw_k\frac{\partial W}{\partial w_k}\log(w_k)\).NEWLINENEWLINEOn the other hand the hyperbolicity equations for the octahedal triangulation turn out to be equivalent to the set of equations \(exp(w_k\frac{\partial W}{\partial w_k})=1\) for \(k=1,\ldots,n\). The paper under review proves that \(W_0\) is locally constant on the solution set of these equations and that it coincides with the complex volume of the boundary-parabolic representation associated to a solution of the hyperbolicity equation.NEWLINENEWLINEIn particular the imaginary part of \(W_0\) for the values \(w_1,\ldots,w_k\) associated to the (unique) complete hyperbolic metric is giving the hyperbolic volume and this reduces the proof of the volume conjecture to a comparison of this ``optimistic limit'' with the classicial limit of the Kashaev invar…
reviewed by: Thilo Kuessner
description: scientific article; zbMATH DE number 6591363