Quantum traces and embeddings of stated skein algebras into quantum tori
Skein modules were introduced by Przytycki and independently by Turaev to describe knot and link invariants of the Kauffman bracket kind on general \(3D\) manifolds. The case of \(3D\) manifolds which are thickened surfaces is particularly fruitful, as it developed into almost a separate domain of mathematics, with connections to Teichmüller spaces, classical and quantum, cluster algebras, noncommutative geometry, quantum groups, and others. It is known that there exist quantum trace maps that embed different realizations of the stated skein algebras in certain quantum tori that are defined using various coordinates in certain generalizations of Teichmüller spaces, and introduced by Bonahon and Wong and by one of the authors. The paper focuses on one hand on generalizing the results to possibly most general \(2D\) surfaces (thickened), with boundaries and punctures, and on the other on finding relationships between the different quantum maps and the embeddings, and there are numerous important results proved in the paper that clarify these relationships.
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