Optimistic limits of the colored Jones polynomials and the complex volumes of hyperbolic links (Q2811982)

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 invariant, though this might still be a hard problem.