|
Organizers |
Does Khovanov homology lead to wedges of spheres?
by
Jozef H. Przytycki
George Washington University
Coauthors: Marithania Silvero (University of Zaragoza)
It has been proven in [GMS] that extreme Khovanov homology is a reduced homology of the simplicial complex obtained from bipartite circle graph constructed from a link diagram (so called Lando graph of a link). In this talk we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, this homotopy type is a link invariant. We prove the conjecture in many special cases and find it convincing to generalize conjecture for any circle graph (intersection graph of cords in a circle). As suggested by M.Adamaszek, we prove that for a permutation graph the conjecture hold. We demonstrate how, for any finite wedge of spheres, to find a permutation graph G with IG of its homotopy type. We modify our construction to show, in particular, that for any n there is a Lando graph G with its independence simplicial complex, IG, homotopy equivalent to Sn∨S2n-1. Another family of graphs, again suggested by Adamaszek, have non-nested cords on one side of the circle. We prove that for this family (and its small generalizations) the conjecture holds. We give several other examples supporting wedge of spheres conjectures, but full conjectures are still open. We would like to thank Michal Adamaszek, Sergei Chmutov, and Victor Reiner for helpful advise.
[GMS] J.González-Meneses, P.M.G.Manchón, M.Silvero, A geometric description of the extreme Khovanov cohomology, e-print: arXiv:1511.05845 [math.GT]
Date received: April 14, 2016
Copyright © 2016 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 # cbmk-29.