TY - JOUR

T1 - Skein modules and the noncommutative torus

AU - Frohman, Charles

AU - Gelca, Räzvan

PY - 2000

Y1 - 2000

N2 - 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 idempotcnts 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.

AB - 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 idempotcnts 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.

KW - Kauffraan bracket

KW - Noncommutative geometry

KW - Skein modules

UR - http://www.scopus.com/inward/record.url?scp=23044522419&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-00-02512-5

DO - 10.1090/s0002-9947-00-02512-5

M3 - Article

AN - SCOPUS:23044522419

SN - 0002-9947

VL - 352

SP - 4877

EP - 4888

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 10

ER -