2016/08/09 by Józef H. Przytycki, Przytycki, Jozef H., Marithania Silvero +1 · 2 citations
Computer Science · Mathematics · Medicine · #57M27 #Botulinum Toxin and Related Neurological Disorders #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Secondary 05E45 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1608.03002
openalex publication_date 2016/08/09 · openalex created_date 2022/09/03 · openalex updated_date 2026/07/28
It was proven by Gonz 'alez-Meneses, Manch 'on and Silvero that the extreme\nKhovanov homology of a link diagram is isomorphic to the reduced (co)homology\nof the independence simplicial complex obtained from a bipartite circle graph\nconstructed from the diagram. In this paper we conjecture that this simplicial\ncomplex is always homotopy equivalent to a wedge of spheres. In particular, its\nhomotopy type, if not contractible, would be a link invariant and it would\nimply that the extreme Khovanov homology of any link diagram does not contain\ntorsion. We prove the conjecture in many special cases and find it convincing\nto generalize it to every circle graph (intersection graph of chords in a\ncircle). In particular, we prove it for the families of cactus, outerplanar,\npermutation and non-nested graphs. Conversely, we also give a method for\nconstructing a permutation graph whose independence simplicial complex is\nhomotopy equivalent to any given finite wedge of spheres. We also present some\ncombinatorial results on the homotopy type of finite simplicial complexes and a\ntheorem generalizing previous results by Csorba, Nagel and Reiner, Jonsson and\nBarmak. We study the implications of our results to Knot Theory; more\nprecisely, we compute the real-extreme Khovanov homology of torus links\nT(3,q) and obtain examples of H-thick knots whose extreme Khovanov homology\ngroups are separated either by one or two gaps as long as desired.\n