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The localized skein algebra is Frobenius

Nel Abdiel and Charles Frohman

Algebraic & Geometric Topology 17 (2017) 3341–3373

When A in the Kauffman bracket skein relation is set equal to a primitive n th 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 SL2–character ring of the fundamental group of F. We localize by inverting the nonzero characters to get an algebra S1Kζ(F) over the function field of the corresponding character variety. We prove that if F is noncompact, the algebra S1Kζ(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.

skein algebra, Frobenius
Mathematical Subject Classification 2010
Primary: 57M27
Received: 11 January 2015
Revised: 11 May 2017
Accepted: 27 May 2017
Published: 4 October 2017
Nel Abdiel
Department of Mathematics
University of Iowa
Iowa City, IA
United States
Charles Frohman
Department of Mathematics
University of Iowa
Iowa City, IA
United States