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

Renaud Detcherry and Ramanujan Santharoubane

Geometry & Topology 29 (2025) 313–348
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 and 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 conjugate to the quaternion group.

Keywords
skein algebras, quantum tori, Azumaya locus
Mathematical Subject Classification
Primary: 57K31
References
Publication
Received: 20 September 2022
Revised: 3 November 2023
Accepted: 4 March 2024
Published: 22 January 2025
Proposed: Stavros Garoufalidis
Seconded: Ciprian Manolescu, András I Stipsicz
Authors
Renaud Detcherry
Institut de Mathématiques de Bourgogne, UMR 5584 CNRS
Université de Bourgogne
Dijon
France
Ramanujan Santharoubane
Laboratoire Mathématiques d’Orsay, UMR 8628 CNRS
Université Paris-Saclay
Orsay
France

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