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ISSN: 0137-6934(p) 1730-6299(e)
 

Frobenius algebras and skein modules of surfaces in 3-manifolds
Uwe Kaiser1
Banach Center Publ. 85 (2009), 59-81
doi:10.4064/bc85-0-4
Abstract: For each (commutative) 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 prove 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 manifold and relations of the algebra.

MSC (2010): Primary 57M25; Secondary 57M35, 57R42.
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  1. Department of Mathematics
    Boise State University
    1910 University Drive
    Boise, ID 83725-1555, U.S.A.