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Abstract: In the 80's, several knot invariants for knots in the 3-dimensional sphere were constructed in a purely combinatorial manner, based on skein relations. Consequently, a study of 3-manifolds in the context of skein modules was initiated, by considering skein relations for knots in arbitrary manifolds. It was noticed that Kauffman bracket skein modules are related to SL(2,C) representations of the fundamental group. In the case when the manifold is a cylinder over a surface, the associated skein module has a natural algebra structure.
The present talk is focused on a joint result with Charles Frohman, in which we show that the Kauffman bracket skein algebra of the cylinder over a torus is canonically isomorphic to the subalgebra of the noncommutative torus spanned by noncommutative cosines. I will describe several applications of this fact to the study of skein modules and to the computation of quantum knot invariants. In particular I will define a noncommutative version of the A-polynomial of a knot, as a finitely generated ideal of polynomials in the quantum plane. |
Geometry-Algebra-Singularities-Combinatorics home page:
http://www.math.neu.edu/~suciu/GASC.html
| Web page: Alexandru I. Suciu | Created: Feb. 5, 1999 Updated: Feb. 5, 1999 |
Comments to:  alexsuciu@neu.edu
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URL: http://www.math.neu.edu/~suciu/gas/gelca.html |