virtual links

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.

Minimal crossing number implies minimal supporting genus

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Will Rushworth (McMaster University)

We prove that a minimal crossing virtual link diagram is a minimal genus diagram. That is, the genus of its Carter surface realizes the supporting genus of the virtual link represented by the diagram. The result is obtained by introducing a new parity theory for virtual links. This answers a basic question in virtual knot theory, and recovers the corresponding result of Manturov in the case of virtual knots.
Joint work with Hans Boden.

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