Optimistic limits of the colored Jones polynomials
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 the \(R\)-matrix with a product of dilogarithms and logarithms, and changing the summations (corresponding to products of \(R\)-matrices) into integrations, one can obtain a (non-rigorous) integral formula for the colored Jones polynomial. Then applying an appropriate saddle point method, one can get the (non-rigorous) limit of the colored Jones polynomial, which is called the optimistic limit [\textit{H. Murakami}, RIMS Kokyuroku 1172, 70--79 (2000; Zbl 0969.57504)], which was inspired by Kashaev's calculation. Thurston's method corresponds to the optimistic limit of \(\log\bigl(J_N(K;\exp(2\pi\sqrt{-1}/N))\bigr)/N\) and Yokota's method corresponds to the optimistic limit of \(\log\langle K\rangle_N/N\).NEWLINENEWLINENote that the optimistic limit depends on the diagram and is not rigorous but that it gives certain supporting evidence to the volume conjecture. It also gives a (rigorous) way to calculate the volume and the Chern--Simons invariant of a hyperbolic knot complement.NEWLINENEWLINEIn the paper under review the authors show that under a certain condition the method by Thurston and that by Yokota give the same optimistic limit for a hyperbolic knot and that they give not only the volume but also the Chern--Simons invariant of the knot complement, which corresponds to the complexification of the volume conjecture [\textit{H. Murakami} et al., Exp. Math. 11, No. 3, 427--435 (2002; Zbl 1117.57300)].
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