Constructions toward topological applications of U(1) x U(1) equivariant Khovanov homology

Melissa Zhang (University of Georgia)

In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still unclear.

Observing that the U(1)xU(1) complex admits two symmetric algebraic gradings, those familiar with knot Floer homology over the ring F[U,V] may naturally ask if these filtrations allow for algebraic constructions already seen in the knot Floer context, such as Ozsváth-Stipsicz-Szabó’s Upsilon. In this talk, I will describe the construction and properties of such an invariant. I will also discuss some ideas on how future research might use the U(1)xU(1) framework to identify invariants similar to those constructed from knot Floer homology over F[U,V], and speculate on the topological information these constructions might illuminate.

This is based on joint work with Ross Akhmechet.

Khovanov Homology for Virtual Links

Ben McCarty (University of Memphis)

Shortly after Khovanov homology for classical knots and links was developed in the late 90s, Bar-Natan produced a Mathematica program for computing it from a planar diagram. Yet when Manturov defined Khovanov homology for virtual links in 2007, a program for computing it did not appear until Tubbenhauer’s 2012 paper. Even then, the theoretical framework used was quite different from the one Manturov described. In this talk, we describe a theoretical framework for computing the Khovanov homology of a virtual link, synthesized from work by Manturov, Dye-Kaestner-Kauffman and others. We also show how to use this framework to create a program for computing the Khovanov homology of a virtual link, which is directly based upon Bar-Natan’s original program for classical links.  This program is joint work with Scott Baldridge, Heather Dye, Aaron Kaestner, and Lou Kauffman.