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Frobenius algebras and skein modules of surfaces in 3-manifolds
by
Uwe Kaiser
Boise State University
For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We discuss a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing geometry in the 3-manifold and relations of the algebra. We discuss some ideas how the structure of the module depends on the 3-manifold (surgery) and how it changes under deformations of the Frobenius algebra (Frobenius manifolds).
Paper reference: arXiv:0802.4068
Date received: April 3, 2008
Copyright © 2008 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 # cawt-27.