When in the Kauffman bracket skein relation is set equal to a primitive root of unity with not divisible by , the Kauffman bracket skein algebra of a finite-type surface is a ring extension of the –character ring of the fundamental group of . We localize by inverting the nonzero characters to get an algebra over the function field of the corresponding character variety. We prove that if is noncompact, the algebra is a symmetric Frobenius algebra. Along the way we prove is finitely generated, 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.
"The localized skein algebra is Frobenius." Algebr. Geom. Topol. 17 (6) 3341 - 3373, 2017. https://doi.org/10.2140/agt.2017.17.3341