Skein modules and framing changes for links in 3-manifolds

Rhea Palak Bakshi (George Washington University)

We show that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating S^2. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of 3-manifolds. We also express our results in the language of skein modules. In particular, we relate our results to the presence of torsion in the framing skein module.

Some algebraic structures for flat virtual links

David Freund (Harvard University)

Flat virtual links are generalizations of curves on surfaces that have come out of Kauffman’s virtual knot theory. In particular, we consider the curves up to homotopy and allow the supporting surface to change via the addition or removal of empty handles. Under this equivalence, a flat virtual link obtains a completely combinatorial structure. Analogous to the classical problem of finding the minimal number of intersection points between two curves, we can ask for the minimal number of intersection points between components of a flat virtual link. By moving between geometric and combinatorial models, we develop generalizations of the Andersen-Mattes-Reshetikhin Poisson bracket and compute it for infinite families of two-component flat virtual links using a generalization of Henrich’s singular based matrix for flat virtual knots. Throughout the talk, we emphasize the motivation behind different constructions.

Diagrammatic Algebra, Part II

J. Scott Carter (University of South Alabama)

In this talk, I discuss replacing axioms in a Frobenius algebra with diagrams and constructing glyphs to represent those diagrams. The ideas are extended to considering isotopy classes of knots as a 4-category. Then we discuss braids, braided manifolds, braid movies, charts, chart movies, curtains, and curtain movies as methods of braiding simple branched covers in codimension 2. As usual, there are lots of diagrams.