Dubois torsion, $A$-polynomial and quantum 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.