Degenerations Of Skein Algebras And Quantum Traces

TL;DR

This paper introduces LRY skein algebras, generalizing existing skein algebras, and explores their algebraic properties, degenerations, and quantum traces, extending results to broader ground rings and surfaces.

Contribution

It defines LRY skein algebras, proves they are domains and Noetherian under certain conditions, and constructs quantum traces and modified coordinates for their study.

Findings

LRY skein algebras are domains with degenerations to quantum tori.

They are Noetherian and finitely generated over certain ground rings.

Quantum traces are constructed for all surfaces, aiding algebraic analysis.

Abstract

We introduce a joint generalization, called LRY skein algebras, of Kauffman bracket skein algebras (of surfaces) that encompasses both Roger-Yang skein algebras and stated skein algebras. We will show that, over an arbitrary ground ring which is a commutative domain, the LRY skein algebras are domains and have degenerations (by filtrations) equal to monomial subalgebras of quantum tori. For surfaces without interior punctures, this integrality generalizes a result of Moon and Wong to the most general ground ring. We also calculate the Gelfand-Kirillov dimension of LRY algebras and show they are Noetherian if the ground ring is. Moreover they are orderly finitely generated. To study the LRY algebras and prove the above-mentioned results, we construct quantum traces, both the so-called X-version for all surfaces and also an A-version for a smaller class of surfaces. We also introduce a modified version of Dehn-Thurston coordinates for curves which are more suitable for the study of skein algebras as they pick up the highest degree terms of products in certain natural filtrations.