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 $\rho:\pi_1(L)\rightarrow{\rm PSL}(2,\mathbb{C})$. This gives us a very simple and combinatorial method to calculate the complex volume of $\rho$.