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12, 359–373 (2003)īarmak, J.A.: Star clusters in independence complexes of graphs. Chmutov, īae, Y., Morton, H.R.: The spread and extreme terms of Jones polynomials. 57, 8–18 (2016)īabson, E.: Personal communication via e-mail to S. We study the implications of our results to knot 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.Īdamaszek, M., Stacho, J.: Complexity of simplicial homology and independence complexes of chordal graphs. 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. Conversely, we also give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge of spheres. In particular, we prove it for the families of cactus, outerplanar, permutation and non-nested graphs. We prove the conjecture in many special cases and find it convincing to generalize it to every circle graph (intersection graph of chords in a circle).
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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. In this paper, we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. 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 constructed from the diagram.