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 $\rho$.
Categories: math.GT