On the Colored Jones Polynomial and the Kashaev invariant

TL;DR

This paper presents a novel determinant formula for the colored Jones polynomial and Kashaev invariant, linking quantum algebra with hyperbolic geometry, and aims to advance understanding of the volume conjecture.

Contribution

It introduces a determinant expression for the colored Jones polynomial and Kashaev invariant using the $q$-Weyl algebra, suggesting a new approach to the volume conjecture.

Findings

Expressed the colored Jones polynomial as a quantum determinant.

Proved the Kashaev invariant equals a special determinant evaluation.

Discussed the connection between the determinant formula and hyperbolic volume.

Abstract

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the $q$-Weyl algebra of $q$-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture.