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


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.


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



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.


  1. 1.
    Adamaszek, M., Stacho, J.: Complexity of simplicial homology and independence complexes of chordal graphs. Comput. Geom. Theory Appl. 57, 8–18 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babson, E.: Personal communication via e-mail to S. Chmutov, 16 Dec 2011Google Scholar
  3. 3.
    Bae, Y., Morton, H.R.: The spread and extreme terms of Jones polynomials. J. Knot Theory Ramif. 12, 359–373 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barmak, J.A.: Star clusters in independence complexes of graphs. Adv. Math. 214, 33–57 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bar-Natan, D., Morrison, S.: The Knot Atlas.
  6. 6.
    Bar-Natan, D.: On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol. 2, 337–370 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Björner, A.: Topological methods. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. II, pp. 1819–1872. North-Holland, Amsterdam (1995). Chapter 34Google Scholar
  8. 8.
    Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fund. Math. 35, 217–234 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bouchet, A.: Circle graph obstructions. J. Comb. Theory Series B 60, 107–144 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brown, R.: Elements of Modern Topology. McGraw Hill, London (1968)zbMATHGoogle Scholar
  11. 11.
    Brown, R.: Topology and Groupoids. Booksurge LLC, Charleston, SC (2006)zbMATHGoogle Scholar
  12. 12.
    Cabello, S., Jejcic, M.: Refining the Hierarchies of Classes of Geometric Intersection Graphs. e-print: arXiv:1603.08974
  13. 13.
    Cannon, J.W.: Shrinking cell-like decompositions of manifolds. Codimension three. Ann. Math. (2) 110(1), 83112 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chartrand, G., Harary, F.: Planar permutation graphs. Ann. Inst. Henri Poincaré 3(4), 433438 (1967)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chmutov, S.: Extreme parts of the Khovanov complex. Abstract of the talk delivered at the conference Knots in Washington XXI: Skein modules, Khovanov homology and Hochschild homology, George Washington University, 9–11 Dec 2005. Notes to the talk are available at
  16. 16.
    Chmutov, S., Duzhin, S., Mostovoy, J.: Introduction to Vassiliev Knot Invariants. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  17. 17.
    Chmutov, S., Lando, S.K.: Mutant knots and intersection graphs. Algebr. Geom. Topol. 7, 1579–1598 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Csorba, P.: Subdivision yields Alexander duality on independence complexes. Electron. J. Comb. 16(2), Research paper 11 (2009)Google Scholar
  19. 19.
    Engström, A.: Complexes of directed trees and independence complexes. Discrete Math. 309(10), 3299–3309 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Everitt, B., Turner, P.: The homotopy theory of Khovanov homology. Algebr. Geom. Topol. 14, 2747–2781 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ghier, L.: Double occurrence words with the same alternance graph. Ars Comb. 36, 57–64 (1993)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Głazek, K.: A Guide to the Literature on Semirings and Their Applications in Mathematics and Information Sciences. With Complete Bibliography. Kluwer Academic Press, Dordrecht (2002)zbMATHGoogle Scholar
  23. 23.
    González-Meneses, J., Manchón, P.M.G., Silvero, M.: A geometric description of the extreme Khovanov cohomology. In: Proceedings of the Royal Society of Edinburgh, Section: A Mathematics (2016, to appear). e-print: arXiv:1511.05845
  24. 24.
    Gorsky, E., Oblomkov, A., Rasmussen, J.: On stable Khovanov homology of torus knots. Exp. Math. 22(3), 265–281 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  26. 26.
    Jonsson, J.: Simplicial complexes of graphs. Lecture Notes on Mathematics. Springer, Berlin (2005)Google Scholar
  27. 27.
    Jonsson, J.: On the topology of independence complexes of triangle-free graphs (2011).
  28. 28.
    Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101, 359–426 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kozlov, D.M.: Complexes of directed trees. J. Comb. Theory Series A 88(1), 112–122 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kronheimer, P.B., Mrowka, T.S.: Khovanov homology is an unknot detector. Publ. Math. l’IHÉS 113, 97–208 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kucharik, M., Hofacker, I., Stadler, P.F., Qin, J.: Pseudoknots in RNA folding landscapes. Bioinformatics 32(2), 187–194 (2016)Google Scholar
  32. 32.
    Lewark, L.: Personal communication, 26 Sept 2016Google Scholar
  33. 33.
    Manchón, P.M.G.: Extreme coeffcients of the Jones polynomial and graph theory. J. Knot Theory Ramif. 13(2), 277–295 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nagel, U., Reiner, V.: Betti numbers of monomial ideals and shifted skew shapes. Electron. J. Comb. 16, 2 (2009)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Przytycki, J. H., Sazdanovic, R.: Torsion in Khovanov homology of semi-adequate links. Fund. Math. 225, 277–303 (2014). e-print: arXiv:1210.5254
  36. 36.
    Shumakovitch, A.: KhoHo—a program for computing and studying Khovanov homology.
  37. 37.
    Stosic, M.: Homological thickness and stability of torus knots. Algebr. Geom. Topol. 7, 261–284 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Traldi, L.: The transition matroid of a 4-regular graph: an introduction. Eur. J. Comb. 50, 180–207 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Viro, O.: Khovanov homology, its definitions and ramifications. Fund. Math. 184, 317–342 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Vernizzi, G., Orland, H., Zee, A.: Prediction of RNA pseudoknots by Monte Carlo simulations. e-print: arXiv:q-bio/0405014
  41. 41.
    Welsh, D.: Complexity: knots, colourings and countings. London Mathematical Society Lecture Note Series no. 186. Cambridge University Press, Cambridge (1993)Google Scholar
  42. 42.
    Wessel, W., Pöschel, R.: On circle graphs. In: Sachs, Horst, Graphs, Hypergraphs and Applications: Proceedings of the Conference on Graph Theory held in Eyba, Teubner-Texte zur Mathematik, vol. 73, p. 207210 (1984)Google Scholar
  43. 43.
    Willis, M.: Stabilization of the Khovanov homotopy type of torus links. Int. Math. Res. Not. 11, 3350–3376 (2017). doi: 10.1093/imrn/rnw127 MathSciNetGoogle Scholar

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

Personalised recommendations