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Homotopy type of circle graph complexes motivated by extreme Khovanov homology

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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.

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Notes

  1. It is worth mentioning here a classical Theorem of Cannon [13], stating that the double suspension of a 3-dimensional homological sphere, e.g., Poincaré sphere, is \(S^5\).

  2. Recall that a semiring is a set X with two binary operations \(+\) and \(\cdot \) and two constants 0 and 1 such that:

    (1) \((X,+,0)\) is a commutative monoid with neutral element 0,

    (2) \((X,\, \cdot \, , 1)\) is a monoid with neutral element 1,

    (3) Multiplication is distributive with respect to addition, that is \((a+b) \cdot c = (a \cdot c) + (b \cdot c)\) and \(c \cdot (a+b) = (c \cdot a) + (c \cdot b)\),

    (4) \(0\cdot a = 0 = a \cdot 0\).

  3. This formula can be interpreted as a lift of a skein relation at the crossing leading to the vertex v (we review the relation vertex-crossing in Sect. 7.2). We do not use this observation in the paper, but one should keep it in mind together with the reflection of Everitt and Turner cited in the Introduction.

  4. Ghier [21] and Chmutov and Lando [17] proved that two chord diagrams have the same circle graph if and only if they are related by a sequence of elementary modifications called mutations (see [38] for historical overview).

  5. In [9] Bouchet gives a complete characterization of circle graphs by combining a minimal set of obstructions with what he calls local complementation operation.

  6. We consider a contractible set to be homotopy equivalent to an empty wedge of spheres.

  7. M. Adamaszek pointed us out the similarity of our Theorem 6.4 with Theorem 3.5 in [4] by Barmak. However, the statement we prove yields additional stronger corollaries than those in [4], e.g., generalized Csorba theorem stated in Corollary 6.9(3).

  8. Given a connected graph G, let \(G_3\) be the graph obtained by replacing each edge of G by a path of length 3. Csorba shows that if G is not a tree and has n vertices and e edges, then \(I_{G_3}\) is homotopy equivalent to \(S^{e-1} \vee S^{n-1}\) [18]. This result follows immediately by using Corollary 6.9(3) \(n-1\) times till one gets the wedge of \(e-n+1\) triangles.

  9. In [23], the Lando graph is defined as the circle graph associated with the state \(s_A\), since that paper deals with the minimal-extreme Khovanov homology (see Remark 7.5).

  10. One can conjecture that for torus knots \(T(p,sp+r)\) with \(p>3\) and fixed \(r < s\), the real-extreme Khovanov homology converges to a finite abelian group when \(s \rightarrow \infty \). With much less confidence we can ask whether in the case \(r=1\) this limit is equal to the real-extreme Khovanov homology of \(T(p,p+1)\). For example, Lukas Lewark checked that for \(T(5,5s+1)\) the real-extreme Khovanov homology \(H^{12s+2,36s+7}T(5,5s+1)=Z_3\), for \(1 \le s \le 8\) [32]. Compare with [24, 37, 43] and with [35, Conjecture 6.1].

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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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Authors and Affiliations

  1. Department of Mathematics, The George Washington University, Washington, DC, USA

    Józef H. Przytycki

  2. University of Gdańsk, Gdańsk, Poland

    Józef H. Przytycki

  3. Departamento de Álgebra, Universidad de Sevilla, Sevilla, Spain

    Marithania Silvero

  4. Institute of Mathematics of the Polish Academy of Science, Warsaw, Poland

    Marithania Silvero

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Correspondence to Józef H. Przytycki.

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Przytycki, J.H., Silvero, M. Homotopy type of circle graph complexes motivated by extreme Khovanov homology. J Algebr Comb 48, 119–156 (2018). https://doi.org/10.1007/s10801-017-0794-y

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