Skein modules and the noncommutative torus

Frohman, Charles; Gelca, Răzvan

doi:10.1090/S0002-9947-00-02512-5

Skein modules and the noncommutative torus
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by Charles Frohman and Răzvan Gelca PDF

Trans. Amer. Math. Soc. 352 (2000), 4877-4888 Request permission

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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