Khovanov homology

Khomotopy type via simplicial complexes and presimplicial sets

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Marithania Silvero (Universidad de Sevilla)

At the end of the past century, Mikhail Khovanov introduced the first homological invariant, now known as Khovanov homology, as a categorification of Jones polynomial. It is a bigraded homology supported in homological and quantum gradings. Given a link diagram, we refer to the maximal (resp. second-to-maximal) quantum grading such that the associated Khovanov complex is non-trivial as extreme (resp. almost extreme) grading.

In this talk we present a new approach to the geometrization of Khovanov homology in terms of simplicial complexes and presimplicial sets, for the extreme and almost-extreme gradings, respectively. We also study the relations of these models with Khovanov spectra, introducec by Robert Lipshitz and Sucharit Sarkar as a spatial refinement of Khovanov homology.

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

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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

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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.

Why unoriented Khovanov homology is useful in graph theory

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Scott Baldridge (LSU)

The Jones polynomial and Khovanov homology of a classical link are oriented link invariants—they depend upon an initial choice of orientation for the link. In this talk, we describe a Jones polynomial and Khovanov homology theory for unoriented virtual links. We then show how to transfer these unoriented versions over to graph theory to construct the Tait polynomial of a trivalent graph. This invariant polynomial counts the number of 3-edge colorings of a graph when evaluated at 1. If this count is nonzero for all bridgeless planar trivalent graphs, then the famous four color theorem is true. Thus, we show how topological ideas can be used to have an impact in graph theory. This is joint work with Lou Kauffman and Ben McCarty.

References:

Baldridge, S., “A cohomology theory for planar trivalent graphs with perfect matchings”, https://arxiv.org/abs/1810.07302

Baldridge, S., Lowrance, A., McCarty, B., “The 2-factor polynomial detects even perfect matchings”, https://arxiv.org/abs/1812.10346

Baldridge, S., Kauffman, L. H., McCarty, B., “Unoriented Khovanov homology”, https://arxiv.org/abs/2001.04512