The localized skein algebra is Frobenius

Abstract

When A in the Kauffman bracket skein relation is set equal to a primitive nth root of unity ζ with n not divisible by 4, the Kauffman bracket skein algebra Kζ(F)of a finite-type surface F is a ring extension of the SL2ℂ-character ring of the fundamental group of F. We localize by inverting the nonzero characters to get an algebra S-1Kζ(F) over the function field of the corresponding character variety. We prove that if F is noncompact, the algebra S-1Kζ(F) is a symmetric Frobenius algebra. Along the way we prove K(F) is finitely generated, Kζ(F) is a finite-rank module over the coordinate ring of the corresponding character variety, and learn to compute the trace that makes the algebra Frobenius.