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Organizers |
Skein modules, quantum deformations and string topology
by
Uwe Kaiser
Siegen University
| Abstract |
We discuss a general approach to understand skein modules of links in an oriented 3-manifold M from the following idea: Skein modules are part of deformed homology theories of suitable multi-loop spaces of M. In general, the multi-loop space is the space of differentiable maps of circles into M. The 0-th homology group of this space is an algebra of free homotopy classes of loops in M (classical observables). This is deformed into the skein modules using the stratification of the multi-loop space by singularities (Vassiliev theory), thus giving the transition to quantum observables. Nontrivial relations for the skein module appear from the string topology of M, e. g. suitable intersection pairings on the multi-loop space.
We discuss several explicit results for skein modules following from this approach: 1.) The universal Jones-Conway invariant (a generalization of the homflypt skein module), 2.) The quantum deformation of the fundamental group, a two-term skein module based on framed oriented links, and 3.) link homotopy skein modules (and their relation with recent work of Chas and Sullivan generalizing the Goldmann-Wolpert Lie algebra).
Date received: May 1, 2001
Copyright © 2001 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cahj-05.