Algebraic generators of the skein algebra of a surface

TL;DR

This paper proves that the skein algebra of certain surfaces can be generated by finitely many simple closed curves linked to the mapping class group, and proposes a conjectural presentation based on these generators.

Contribution

It establishes a finite generating set for the skein algebra of surfaces with negative Euler characteristic and genus at least one, connected to the mapping class group actions.

Findings

Finite generating set for the skein algebra.

Relations derived from the mapping class group actions.

Conjectured presentation for the skein algebra.

Abstract

Let $\Sigma$ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the skein algebra of $\Sigma$ over the field of rational functions can be algebraically generated by a finite number of simple closed curves that are naturally associated to certain generators of the mapping class group of $\Sigma$. The action of the mapping class group on the skein algebra gives canonical relations between these generators. From this, we conjecture a presentation for a skein algebra of $\Sigma$.