A splitting property of the chromatic homology of the complete graph
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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2023
- DOI:
- 10.48550/arXiv.2301.01972
- arXiv:
- arXiv:2301.01972
- Bibcode:
- 2023arXiv230101972Y
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Algebraic Topology;
- Mathematics - Combinatorics;
- 57M15;
- 05C15
- E-Print:
- 20 pages, 5 figures