A Spanning Tree Model for Chromatic Homology

In this talk, I will talk about chromatic homology, a categorification of the chromatic polynomial of a graph. Algebraic Morse theory is a powerful technique that simplifies chain complexes by reducing the number of generators through an acyclic matching on their Hasse diagrams. I will show that, using an acyclic matching, the chromatic complex can be generated solely by its spanning trees. This provides a categorified version of the spanning tree expansion formula of the chromatic polynomial.

I will then discuss how this model offers new proofs of several properties of the chromatic homology that were previously established using algebraic methods. Additionally, it helps resolve certain conjectures about the algebraic structure of chromatic homology. This talk is based on joint work with Aninda Banerjee, Swarup Das, and Pravakar Paul.