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

Abstract

In earlier work, the authors introduced a conjecture which, for an orientation-preserving diffeomorphism $\phi :S\to S$ of a surface, connects a certain quantum invariant of $\phi$ with the hyperbolic volume of its mapping torus $M_{\phi }$. 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 $M_{\phi }$ is the complement of the figure-eight knot.

30 pages, 3 figures