Frobenius algebras and skein modules of surfaces in 3-manifolds
Tom 85 / 2009
Banach Center Publications 85 (2009), 59-81
MSC: Primary 57M25; Secondary 57M35, 57R42.
DOI: 10.4064/bc85-0-4
Streszczenie
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.
