On the Potential Function of the Colored Jones Polynomial with Arbitrary Colors

TL;DR

This paper explores the potential function of the colored Jones polynomial for links with arbitrary colors, linking saddle point equations to hyperbolic structures and supporting the Chen-Yang conjecture.

Contribution

It establishes a connection between the potential function, hyperbolic geometry, and the Chen-Yang conjecture for colored Jones polynomials of links.

Findings

Derived the cone-manifold structure for link complements.

Linked saddle point equations to hyperbolicity.

Provided evidence supporting the Chen-Yang conjecture.

Abstract

We consider the potential function of the colored Jones polynomial for a link with arbitrary colors and obtain the cone-manifold structure for the link complement. In addition, we establish a relationship between a saddle point equation and hyperbolicity of the link complement. This provides evidence for the Chen-Yang conjecture on the link complement.