2023/01/05 by So Yamagata, Yamagata, So
Computer Science · Mathematics · #05C15 #57M15 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2301.01972
openalex publication_date 2023/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.