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State sums, mapping classes and bracket deformation


Speaker: Uwe Kaiser, Boise State University
Location: 2112 MSB
Start time: Wed, May 17 2006, 4:10PM

The Kauffman bracket algebra of an oriented surface is a free module with basis the isotopy classes of systems of disjoint essential curves on the surface. It is known that the natural multiplication of the algebra deforms the commutative product of the SL(2,C)-characters of the fundamental group of the surface. Moreover the mapping class group acts on the bracket algebra in a natural way. The module isomorphism is given by the usual Kauffman state sum. I will describe a formula, which expands this state sum in terms of a natural algebra of diagram resolutions according to the order of deformation. This allows to prove some new results about the nature of the deformation. Moreover I will give some background and examples for the problem of explicit calculation of the product of two essential curves in the bracket module. This is joint work with Nikos Apostolakis