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Khovanov homology for virtual links with two types of maps for Möbius cobordisms
by
Kokoro Tanaka
Tokyo Gakugei University
Coauthors: Atsushi Ishii (University of Tsukuba)
Khovanov homology is a homology theory for (classical) links which is a categorification of the Jones polynomial. If we want to extend Khovanov homology to virtual links, Khovanov's construction does not immediately work and the main difficulty arising is the existence of Möbius cobordisms (bifurcations of type 1 → 1). Recently, V. O. Manturov succeeded in extending Khovanov homology to virtual links. In his construction, the zero map is assigned to each of the Möbius cobordisms because of the grading reasons.
In this talk, we construct a new extension of Khovanov homology to virtual links by taking suitable grading shifts derived from the Miyazawa polynomial, which is known as a multi-variable generalization of the Jones-Kauffman polynomial. In our construction, we assign one of two non-zero maps to each of the Möbius cobordisms.
Date received: December 1, 2008
Copyright © 2008 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caxq-45.