Kauffman bracket intertwiners and the volume conjecture

TL;DR

This paper investigates the intertwiners in the skein algebra related to the volume conjecture, providing explicit calculations for the torus and supporting the conjecture for punctured tori through algebraic and geometric analysis.

Contribution

It explicitly computes intertwiners for the closed torus and verifies the conjecture for the once punctured torus, advancing understanding of the volume conjecture in quantum topology.

Findings

Limit superior of trace of intertwiners is zero for the closed torus.

Conjecture that the limit is zero for surfaces with negative Euler characteristic.

Confirmed the conjecture for the once punctured torus.

Abstract

The volume conjecture relates the quantum invariant and the hyperbolic geometry. Bonahon-Wong-Yang introduced a new version of the volume conjecture by using the intertwiners between two isomorphic irreducible representations of the skein algebra. The intertwiners are built from surface diffeomorphisms; they formulated the volume conjecture when diffeomorphisms are pseudo-Anosov. In this paper, we explicitly calculate all the intertwiners for the closed torus using an algebraic embedding from the skein algebra of the closed torus to a quantum torus, and show the limit superior related to the trace of these intertwiners is zero. Moreover, we consider the periodic diffeomorphisms for surfaces with negative Euler characteristic, and conjecture the corresponding limit is zero because the simplicial volume of the mapping tori for periodic diffeomorphisms is zero. For the once punctured torus, we make precise calculations for intertwiners and prove our conjecture.