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Skein modules and the noncommutative torus
Author(s):
Charles
Frohman;
Razvan
Gelca
Journal:
Trans. Amer. Math. Soc.
352
(2000),
4877-4888.
MSC (1991):
Primary 57M27, 58B32, 47L87
Posted:
June 12, 2000
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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  -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:
10.1090/S0002-9947-00-02512-5
PII:
S 0002-9947(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.
Posted:
June 12, 2000
Copyright of article:
Copyright
2000,
American Mathematical Society
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