A splitting property of the chromatic homology

So Yamagata

Published 2023-01-05Version 1

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 \cite{HY}, resulting in a categorification known as the ``chromatic homology''. Much as in the Khovanov homology, in the chromatic 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. Finally, as an application of the description, we compute the chromatic homology of the complete graph.