Why unoriented Khovanov homology is useful in graph theory

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