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Virtual Khovanov homology using cobordisms

Daniel Tubbenhauer
Organization: Centre for Quantum Geometry of Moduli Spaces (QGM)
Department: Department of Mathematics
Journal / Anthology

Journal of Knot Theory and Its Ramifications
Year: 2014
Volume: 23
Issue: 9

We extend Bar-Natanís cobordism-based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h = t = 0), the variant of Lee (h = 0, t = 1) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natanís Z/2-link homology can be extended in two nonequivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICAbased program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semilocal properties to prove an application, i.e. we give a discussion of Leeís degeneration of virtual homology.

*Applied Mathematics