Kauffman bracket intertwiners and the volume conjecture

Wang, Zhihao

doi:10.2140/agt.2025.25.2143

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Kauffman bracket intertwiners and the volume conjecture

Zhihao Wang

Algebraic & Geometric Topology 25 (2025) 2143–2177

DOI: 10.2140/agt.2025.25.2143

Abstract

The volume conjecture relates the quantum invariant and the hyperbolic geometry. Bonahon, Wong and 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. 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.

Keywords

volume conjecture, skein algebra, representation theory

Mathematical Subject Classification

Primary: 14H10, 14H30, 14H45, 14H50, 14L10

References

Publication

Received: 9 February 2023

Revised: 15 February 2024

Accepted: 18 May 2024

Published: 11 August 2025

Authors

Zhihao Wang
Bernoulli Institute University of Groningen Groningen Netherlands

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Volume 25, issue 4 (2025)