Torsion in graph homology
Volume 190 / 2006
Fundamenta Mathematicae 190 (2006), 139-177
MSC: Primary 57M27; Secondary 05C15, 55N35.
DOI: 10.4064/fm190-0-5
Abstract
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 torsion. When the underlying algebra is $\mathbb{Z}[x]/(x^2)$, we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov–Rozansky $sl(m)$ homology of knots (in particular $(2,n)$ torus knot). We also predict that our work is connected with Hochschild and Connes cyclic homology of algebras.
