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On the first group of the chromatic cohomology of graphs

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Abstract

The Hochschild homology of the algebra of truncated polynomials \({{\mathcal {A}_m=\mathbb {Z}[x]/(x^m)}}\) is closely related to the Khovanov-type homology as shown by the second author. In the present paper we utilize this in the study of the first graph cohomology group of an arbitrary graph G with v vertices. The complete description of this group is given for m = 2, 3. For the algebra \({\mathcal {A}_2}\) we relate the chromatic graph cohomology with the Khovanov homology of adequate links. We describe the chromatic cohomology over the algebra \({\mathcal {A}_3}\) using the homology of a cell complex built on the graph G. In particular we prove that \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) can be isomorphic to any finite abelian group. Moreover, we give a characterization of graphs which have torsion in cohomology \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) and construct graphs which have the same (di)chromatic polynomial but different \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) .

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Authors and Affiliations

  1. Department of Mathematics, The George Washington University, Washington, DC, 20052, USA

    Milena D. Pabiniak, Józef H. Przytycki & Radmila Sazdanović

  2. University of Maryland, College Park, USA

    Józef H. Przytycki

  3. 10005 Broad Street, Bethesda, MD, 20814, USA

    Józef H. Przytycki

Authors
  1. Milena D. Pabiniak
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  2. Józef H. Przytycki
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  3. Radmila Sazdanović
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Correspondence to Józef H. Przytycki.

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Pabiniak, M.D., Przytycki, J.H. & Sazdanović, R. On the first group of the chromatic cohomology of graphs. Geom Dedicata 140, 19–48 (2009). https://doi.org/10.1007/s10711-008-9307-4

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  • DOI: https://doi.org/10.1007/s10711-008-9307-4

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