DIMENSION AND TRACE OF THE KAUFFMAN BRACKET SKEIN ALGEBRA

Abstract

Let F be a finite type surface and ζ a complex root of unity. The Kauffman bracket skein algebra Kζ (F) is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of Kζ (F) over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of F .