Oleg Viro
Published 2006-11-13, updated 2006-11-25Version 2
By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov homology; alternatable virtual links intermediate between virtual and classical links. The most profound role here belongs to a structure that we dare to call orientation of chord diagram. Khovanov homology is generalized to oriented signed chord diagrams and links in oriented thickened surface such that the link projection realizes the first Stiefel-Whitney class of the surface. After this paper was published, V.O.Manturov succeeded in extending Khovanov homology with arbitrary coefficients to arbitrary virtual links, see arXiv: math.GT/0601152.
Comments: Added references and post-publication remarks
Journal: 2005 Gokova Geometry/Topology Conference Proceedings, 2006, pp. 184 - 209
A Turaev surface approach to Khovanov homology
On odd torsion in even Khovanov homology
A generalized skein relation for Khovanov homology and a categorification of the $θ$-invariant
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