Dubois' Torsion, A-polynomial and Quantum Invariants

TL;DR

This paper demonstrates that Dubois' torsion can detect the A-polynomial of knots with regular character varieties and provides a global integral formula linking quantum invariants and classical knot invariants.

Contribution

It establishes a connection between Dubois' torsion, the A-polynomial, and quantum invariants, offering a new perspective on their interplay in knot theory.

Findings

Dubois' torsion detects the A-polynomial for certain knots.

A global integral formula for Dubois' torsion is derived.

The formula resembles heat kernel regularization of Witten-Reshetikhin-Turaev invariants.

Abstract

It is shown that for knots with a sufficiently regular character variety the Dubois' torsion detects the A-polynomial of the knot. A global formula for the integral of the Dubois torsion is given. The formula looks like the heat kernel regularization of the formula for the Witten-Reshetikhin-Turaev invariant of the double of the knot complement. The Dubois' torsion is recognized as the pushforward of a measure on the character variety of the double of the knot complement coming from the square root of Reidemeister torsion. This is used to motivate a conjecture about quantum invariants detecting the A-polynomial.