Torsion in Graph Homology

Details Torsion in Graph Homology Torsion in Graph Homology

2005-07-12

arxiv.org/abs/math/0507245v2

Mathematics

Geometric Topology

45 pages, 20 figures, Fundamenta Mathematicae, 190, 2006, to appear

Scientific paper

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is $\mathbb{Z}[x]/(x^2)$, we determine precisely those graphs whose cohomology contains torsion. For a larger class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. The ideas of this paper could potentially be used to predict the Khovanov-Rozansky $sl(m)$ homology of knots (in particular $(2,n)$ torus knots). We also predict that our work is connected with Hochschild and Connes cyclic homology of algebras.

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