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Journal of Algebraic Combinatorics

, Volume 48, Issue 1, pp 119–156 | Cite as

Homotopy type of circle graph complexes motivated by extreme Khovanov homology

  • Józef H. PrzytyckiEmail author
  • Marithania Silvero
Article

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 constructed from 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 generalize it 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 of spheres. 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 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.

Keywords

Circle graphs Independence simplicial complex Khovanov homology Torus links Wedge of spheres 

Notes

Acknowledgements

J. H. Przytycki was partially supported by Simons Collaboration Grant-316446, and M. Silvero was partially supported by MTM2013-44233-P and FEDER. We would like to thank Michał Adamaszek and Victor Reiner for many useful discussions. In particular, Reiner helped us with the original version of Sect. 2.2. The authors are grateful to the Institute of Mathematics of the University of Seville (IMUS) and the Institute of Mathematics of the University of Barcelona (IMUB) for their hospitality.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe George Washington UniversityWashingtonUSA
  2. 2.University of GdańskGdańskPoland
  3. 3.Departamento de ÁlgebraUniversidad de SevillaSevillaSpain
  4. 4.Institute of Mathematics of the Polish Academy of ScienceWarsawPoland

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