Kauffman bracket intertwiners and the volume conjecture

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.

35 pages, the update for the accepted version, to appear in Algebraic & Geometric Topology

References in corpus (4)

• Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations
• Representations of the Kauffman bracket skein algebra II: punctured surfaces
• Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality
• The Witten-Reshetikhin-Turaev representation of the Kauffman skein algebra