Abstract
In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. Helme-Guizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only non-trivial cohomology groups Hi, n - i (Γ), Hi, n - i - 1 (Γ) come in isomorphic pairs: Hi, n - i (Γ) ≅ Hi + 1, n - i - 2 (Γ) for i ≥ 0 if Γ is non-bipartite, and for i > 0 if Γ is bipartite. As a corollary, the ranks of the cohomology groups are determined by the chromatic polynomial. At the end, we give an explicit formula for the Poincaré polynomial in terms of the chromatic polynomial and a deletion-contraction formula for the Poincaré polynomial.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 311-321 |
| Number of pages | 11 |
| Journal | European Journal of Combinatorics |
| Volume | 29 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2008 |
Bibliographical note
Funding Information:This work was motivated by computer calculations of the chromatic homology by M. Chmutov, which revealed certain patterns in the Betti numbers. Part of this work was completed during the Summer’05 VIGRE working group “Knot Theory and Combinatorics” at the Ohio State University funded by NSF, grant DMS-0135308. Y. Rong was partially supported by the NSF grant DMS-0513918. The authors would like to thank L. Helme-Guizon and J. Przytycki for numerous discussions, S. Duzhin and anonymous referees for valuable comments.