On the structure of Kauffman bracket skein algebra of a surface

TL;DR

This paper investigates the algebraic structure of the Kauffman bracket skein algebra of a surface, providing generators and relations with bounded degree, under specific conditions on embedded graphs.

Contribution

It introduces a method to generate the skein algebra using embedded graphs and describes the relations with degree at most 6, under mild conditions.

Findings

Generators are associated with certain embedded graphs.

Relations are generated by relations of degree at most 6.

The structure is characterized for surfaces with specific graph conditions.

Abstract

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of $\Sigma$ over $R$. It is shown that to each embedded graph $G\subset\Sigma$ satisfying that $\Sigma\setminus G$ is homeomorphic to a disk and some other mild conditions, one can associate a generating set for $\mathcal{S}(\Sigma;R)$, and the ideal of defining relations is generated by relations of degree at most $6$ supported by certain small subsurfaces.