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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc85-0-4,
author = {Uwe Kaiser},
title = {Frobenius algebras and skein modules of surfaces in 3-manifolds},
journal = {Banach Center Publications},
volume = {86},
year = {2009},
pages = {59-81},
zbl = {1181.57008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc85-0-4}
}
Uwe Kaiser. Frobenius algebras and skein modules of surfaces in 3-manifolds. Banach Center Publications, Tome 86 (2009) pp. 59-81. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc85-0-4/