## Homotopy type of circle graph complexes motivated by extreme Khovanov homology

### Józef Henryk Przytycki , Marithania Silvero

#### Abstract

It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructedfrom the diagram. In this paper, we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant (up to suspension), and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion. We prove the conjecture in many special cases and find it convincing to generalizeit to every circle graph (intersection graph of chords in a circle). In particular, we prove it for the families of cactus, outerplanar, permutation and non-nested graphs.Conversely, we also give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge ofspheres. We also present some combinatorial results on the homotopy type of finite simplicial complexes and a theorem shedding light on previous results by Csorba, Nagel and Reiner, Jonsson and Barmak. We study the implications of our results toknot theory; more precisely, we compute the real-extreme Khovanov homology of torus links T (3, q) and obtain examples of H-thick knots whose extreme Khovanov homology groups are separated either by one or two gaps as long as desired.
 Author Józef Henryk Przytycki (FMPI / IM) Józef Henryk Przytycki,, - Institute of Mathematics , Marithania Silvero Marithania Silvero,, - Journal series Journal of Algebraic Combinatorics, ISSN 0925-9899, (A 30 pkt) Issue year 2018 Vol 48 No 1 Pages 119-156 Publication size in sheets 1.85 Keywords in English circle graphs, independence simplicial complex, Khovanov homology, torus links, wedge of spheres DOI DOI:10.1007/s10801-017-0794-y URL https://link.springer.com/content/pdf/10.1007%2Fs10801-017-0794-y.pdf Language en angielski Score (nominal) 30 Score Ministerial score = 30.0, 13-11-2018, ArticleFromJournalMinisterial score (2013-2016) = 30.0, 13-11-2018, ArticleFromJournal Publication indicators WoS Impact Factor: 2017 = 0.73 (2) - 2017=0.806 (5) Citation count*
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