Asymptotics of quantum invariants of surface diffeomorphisms II: The figure-eight knot complement

TL;DR

This paper proves a conjecture linking quantum invariants of surface diffeomorphisms to hyperbolic volume in the specific case of the one-puncture torus and the figure-eight knot complement.

Contribution

It provides a proof of a conjecture connecting quantum invariants and hyperbolic volume for the figure-eight knot complement case.

Findings

Confirmed the conjecture for the one-puncture torus case

Established the relationship between quantum invariants and hyperbolic volume

Enhanced understanding of quantum invariants in 3-manifold topology

Abstract

In earlier work, the authors introduced a conjecture which, for an orientation-preserving diffeomorphism $\varphi \colon S \to S$ of a surface, connects a certain quantum invariant of $\varphi$ with the hyperbolic volume of its mapping torus $M_\varphi$. This article provides a proof of this conjecture in the simplest case where it applies, namely when the surface $S$ is the one-puncture torus and the mapping torus $M_\varphi$ is the complement of the figure-eight knot.