Optimistic limit of the colored Jones polynomial and the existence of a solution

Abstract

For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic representation $\rho :{\pi }_1(L)\to \mathrm{PSL}(2,\mathbb{C})$. This gives us a very simple and combinatorial method to calculate the complex volume of $\rho$.

Updated explanations. 14 pages, 9 figures. To appear in Proc. Amer. Math. Soc