Mahler measure of the colored Jones polynomial and the volume conjecture

TL;DR

This paper explores the potential link between the Mahler measure of the colored Jones polynomial and the volume conjecture, focusing on the figure-eight knot and proposing a method to prove the conjecture for its satellites.

Contribution

It introduces a possible relation between Mahler measure and the volume conjecture and proposes a new approach to prove the conjecture for satellite knots of the figure-eight.

Findings

Analysis of the Mahler measure of the colored Jones polynomial for the figure-eight knot.

Proposal of a method to prove the volume conjecture for satellites of the figure-eight knot.

Discussion of the relation between Mahler measure and the volume conjecture.

Abstract

In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I will also propose a method to prove the volume conjecture for satellites of the figure-eight knot.