Dimension and Trace of the Kauffman Bracket Skein Algebra

Abstract

Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The Kauffman bracket skein algebra $K_{\zeta}(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichm\"uller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_{\zeta}(F)$ over its center, and we extend a theorem of Frohman and Kania-Bartoszynska which says the skein algebra has a splitting coming from two pants decompositions of $F$.