On the homology theory for the chromatic polynomials

Zipei Zhuang

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arXiv · 2107.03671

On the homology theory for the chromatic polynomials

Zipei Zhuang

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Abstract

In \cite{10.2140/agt.2005.5.1365}, Rong and Helme-Guizon defined a categorification for the chromatic polynomial $P_G(x)$ of graphs $G$, i.e. a homology theory $H^*(G)$ whose Euler characteristic equals $P_G(x)$. In this paper, we showed that the rational homoology $H^*(G;\mathbb{Q})$ is supported in two lines, and develop an analogy of Lee's theory for Khovanov homology. In particular, we develop a new homology theory $H_{Lee}(G)$, and showed that there is a spectral sequence whose $E_2$ -term is isomorphic to $H^*(G)$ converges to $H_{Lee}(G)$.

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Zipei Zhuang. 2021-07-08. On the homology theory for the chromatic polynomials. https://arxiv.org/abs/2107.03671

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