On the structure of Kauffman bracket skein algebra of a surface

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.