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

TL;DR

This paper demonstrates a direct combinatorial method to find solutions to hyperbolicity equations derived from the optimistic limit of the colored Jones polynomial, enabling calculation of complex volumes associated with boundary-parabolic representations.

Contribution

It introduces a new constructive approach linking shadow-coloring and hyperbolicity equations to compute complex volumes of link complements.

Findings

Existence of solutions to hyperbolicity equations established

Method simplifies calculation of complex volumes

Connects colored Jones polynomial with hyperbolic geometry

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 $\rho$.