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 ρ : π 1 (L) → PSL(2, C). This gives us a very simple and combinatorial method to calculate the complex volume of ρ.
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