Quantum Topology Seminar
Louis H Kauffman
UIC and NSU
Trivalent Graphs and Virtual Links
Abstract: This talk is joint work with Scott Baldridge and Willam Rushworth.
We define a correspondence between trivalent virtual graphs (trivalent ribbon graphs) and virtual link diagrams (abstract link diagrams) so that it is seen that
a generalization of the Penrose evaluation for three-coloring trivalent graphs corresponds to the Kauffman bracket polynomial. The generalization of the Penrose evaluation
is a polynomial depending on a perfect matching in the graph. Thus a graph with a perfect matching corresponds to a virtual link.
This leads to an interaction between graph theory and virtual link
theory that allows us to examine many invariants across this relationship and to define integral Khovanov homology for trivalent graphs with perfect matchings.
The definition of Khovanov homology that we discuss is a new simplification of the integral Khovanov homology for virtuals originally defined by Manturov and further studied by
Dye, Kaestner and Kauffman. The new version is also studied by Kauffman and Ogasa and by Baldridge,Kauffman and McCarty.
Remark. I gave this talk to a group at GWU last Friday and I have spoken about this topic before in the Quantum Topology Seminar. This talk will review the basics and then discuss some work on finding
strong embeddings of graphs that is related to the functor from trivalent graphs to virtual links.
Thursday October 8, 2020 at 4:00 PM in Zoom