Algebraic & Geometric Topology

The localized skein algebra is Frobenius

Nel Abdiel and Charles Frohman

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 6 (2017), 3341-3373.

Received: 11 January 2015
Revised: 11 May 2017
Accepted: 27 May 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds

skein algebra Frobenius


Abdiel, Nel; Frohman, Charles. The localized skein algebra is Frobenius. Algebr. Geom. Topol. 17 (2017), no. 6, 3341--3373. doi:10.2140/agt.2017.17.3341.

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