Reidemeister transformations of the potential function and the solution

TL;DR

This paper explores how Reidemeister moves affect the potential function and solutions related to hyperbolic link invariants, providing explicit formulas to track changes in complex volume and link representations.

Contribution

It introduces explicit formulas for Reidemeister transformations of the potential function and solutions, linking diagram moves to hyperbolic invariants and representations.

Findings

Formulas for Reidemeister transformations of the potential function

Explicit description of changes in complex volume under Reidemeister moves

Method to specify discrete faithful representations from diagrams

Abstract

The potential function of the optimistic limit of the colored Jones polynomial and the construction of the solution of the hyperbolicity equations were defined in the authors' previous articles. In this article, we define the Reidemeister transformations of the potential function and the solution by the changes of them under the Reidemeister moves of the link diagram and show the explicit formulas. These two formulas enable us to see the changes of the complex volume formula under the Reidemeister moves. As an application, we can simply specify the discrete faithful representation of the link group by showing a link diagram and one geometric solution.