Asymptotics of quantum invariants of surface diffeomorphisms I: conjecture and algebraic computations

TL;DR

This paper proposes a new conjecture relating surface diffeomorphisms to quantum invariants, supported by numerical evidence and algebraic computations, extending ideas from knot theory to surface mappings.

Contribution

It introduces a conjecture connecting surface diffeomorphisms with quantum invariants and develops explicit algebraic methods to compute related isomorphisms.

Findings

Numerical evidence supports the conjecture.

Developed explicit algebraic techniques for computations.

Proved the conjecture for a large family of diffeomorphisms in subsequent work.

Abstract

The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We introduce a closely related conjecture for diffeomorphisms of surfaces, backed up by numerical evidence. The conjecture involves isomorphisms between certain representations of the Kauffman bracket skein algebra of the surface, and the bulk of the article is devoted to the development of explicit methods to compute these isomorphisms. These combinatorial and algebraic techniques are exploited in two subsequent articles, which prove the conjecture for a large family of diffeomorphisms of the one-puncture torus and are much more analytic.