On the first group of the chromatic cohomology of graphs

TL;DR

This paper explores the chromatic cohomology of graphs related to algebraic structures used in link homology theories, revealing new torsion phenomena and characterizations of graphs with specific cohomological properties.

Contribution

It provides explicit calculations and characterizations of graph cohomology groups associated with A_m algebras, highlighting torsion occurrences and their relation to graph features.

Findings

Hochschild homology relates to Khovanov homology via graph cohomology.

Torsion appears in specific gradings of Hochschild homology of A_m.

Graphs with certain cycles have torsion in their cohomology.

Abstract

The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph cohomology. It is not difficult to compute Hochschild homology of A_m and the only torsion, equal to Z_m, appears in gradings (i,m(i+1)/2) for any positive odd i. We analyze here the grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H^{1,v-1}_{A_2}(G) and H^{1,2v-3}_{A_3}(G). The group H^{1,v-1}_{A_2}(G) is closely related to the standard graph cohomology, except that the boundary of an edge is the sum of endpoints instead of the difference. The result about H^{1,v-1}_{A_2}(G) gives as a corollary a fact about Khovanov homology of alternating and + or - adequate link diagrams. The group H^{1,2v-3}_{A_3}(G) can be computed from the homology of a cell complex, X_{\Delta,4}(G), built from the graph G. In particular, we prove that A_3 cohomology can have any torsion. We give a simple and complete characterization of those graphs which have torsion in cohomology H^{1,2v-3}_{A_3}(G) (e.g. loopless graphs which have a 3-cycle). We also construct graphs which have the same (di)chromatic polynomial but different H^{1,2v-3}_{A_3}(G). Finally, we give examples of calculations of width of H^{1,*}_{A_3}(G) and of cohomology H^{1,(m-1)(v-2)+1}_{A_m}(G) for m>3.