On the first group of the chromatic cohomology of graphs

الأصل · EN

The algebra of truncated polynomials Aₘ=Z[x]/(xᵐ) 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 Aₘ and the only torsion, equal to Zₘ, 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 H¹,ᵛ⁻¹ₐ₂(G) and H¹,²ᵛ⁻³ₐ₃(G). The group H¹,ᵛ⁻¹ₐ₂(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 H¹,ᵛ⁻¹ₐ₂(G) gives as a corollary a fact about Khovanov homology of alternating and + or - adequate link diagrams. The group H¹,²ᵛ⁻³ₐ₃(G) can be computed from the homology of a cell complex, XΔ,₄(G), built from the graph G. In particular, we prove that A₃ cohomology can have any torsion. We give a simple and complete characterization of those graphs which have torsion in cohomology H¹,²ᵛ⁻³ₐ₃(G) (e.g. loopless graphs which have a 3-cycle). We also construct graphs which have the same (di)chromatic polynomial but different H¹,²ᵛ⁻³ₐ₃(G). Finally, we give examples of calculations of width of H¹,*ₐ₃(G) and of cohomology H¹,⁽ᵐ⁻¹⁾⁽ᵛ⁻²⁾⁺¹ₐₘ(G) for m>3.