Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Skein modules and the noncommutative torus


Authors: Charles Frohman and Razvan Gelca
Journal: Trans. Amer. Math. Soc. 352 (2000), 4877-4888
MSC (1991): Primary 57M27, 58B32, 47L87
Published electronically: June 12, 2000
MathSciNet review: 1675190
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Abstract:

We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the $n$-th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.


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Additional Information

Charles Frohman
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: frohman@math.uiowa.edu

Razvan Gelca
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, and Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Email: rgelca@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02512-5
Keywords: Kauffman bracket, skein modules, noncommutative geometry
Received by editor(s): June 15, 1998
Received by editor(s) in revised form: January 20, 1999
Published electronically: June 12, 2000
Article copyright: © Copyright 2000 American Mathematical Society