A splitting property of the chromatic homology of the complete graph

TL;DR

This paper introduces a combinatorial approach to analyze the chromatic homology of graphs, proves a splitting property for certain graphs, and explicitly computes the chromatic homology of the complete graph.

Contribution

It provides a new combinatorial description of enhanced states and demonstrates a splitting property of chromatic homology for specific graphs, including the complete graph.

Findings

Established a splitting property of chromatic homology for a class of graphs.

Developed a combinatorial method for analyzing chromatic homology.

Computed the chromatic homology of the complete graph explicitly.

Abstract

Khovanov introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones polynomial. The theory was subsequently applied to the chromatic polynomial of graph, resulting in a categorification known as the ``chromatic homology''. Much as in the Khovanov homology, the chromatic polynomial can be obtained by taking the Euler characteristic of the chromatic homology. In the present paper, we introduce a combinatorial description of enhanced states that can be applied to analysis of the homology in an explicit way by hand. Using the new combinatorial description, we show a splitting property of the chromatic homology for a certain class of graphs. Finally, as an application of the description, we compute the chromatic homology of the complete graph.