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Frobenius algebras and skein modules of surfaces in 3-manifolds

Volume 85 / 2009

Uwe Kaiser Banach Center Publications 85 (2009), 59-81 MSC: Primary 57M25; Secondary 57M35, 57R42. 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.

Authors

  • Uwe KaiserDepartment of Mathematics
    Boise State University
    1910 University Drive
    Boise, ID 83725-1555, U.S.A.
    e-mail

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