The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of the figure-eight knot

TL;DR

This paper demonstrates how the asymptotic analysis of the colored Jones polynomial for the figure-eight knot can reveal geometric invariants like the Chern--Simons invariant and Reidemeister torsion, linking quantum invariants to classical topology.

Contribution

It introduces a method to extract classical geometric invariants from the asymptotic behavior of quantum knot invariants.

Findings

Extraction of the Chern--Simons invariant from polynomial asymptotics

Recovery of the Reidemeister torsion via asymptotic evaluation

Establishes a connection between quantum invariants and classical topology

Abstract

We show that from the asymptotic behavior of an evaluation of the colored Jones polynomial of the figure-eight knot we can extract the Chern--Simons invariant and the twisted Reidemeister torsion associated with a representation of the fundamental group of the knot complement to the two-dimensional complex special linear group.