An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates

TL;DR

This paper constructs explicit embeddings of skein algebras of surfaces into quantum tori, enabling detailed study of their representations at roots of unity and providing a new proof of a key conjecture.

Contribution

It introduces a novel embedding method of skein algebras into quantum tori, facilitating analysis of their representations and proving the Bonahon-Wong unicity conjecture.

Findings

Explicit embeddings into quantum tori are constructed.

The method allows reconstruction of unique representations with fixed classical shadows.

A new proof of Bonahon-Wong's unicity conjecture is provided.

Abstract

We construct embeddings of Kauffman bracket skein algebras of surfaces (either closed or with boundary) into localized quantum tori using the action of the skein algebra on the skein module of the handlebody. We use those embeddings to study representations of Kauffman skein algebras at roots of unity and get a new proof of Bonahon-Wong's unicity conjecture. Our method allows one to explicitly reconstruct the unique representation with fixed classical shadow, as long as the classical shadow is irreducible with image not conjuguate to the quaternion group.