Homotopy type of circle graph complexes motivated by extreme Khovanov homology

Abstract

It was proven in [GMS] 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 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 generalizing 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.

Extreme Khovanov spectra

“…, n − 1, such that X 0 is contractible, Y i is a wedge of spheres and X n = X . In [PS18] it was conjectured that |X D | is homotopy equivalent to a wedge of spheres for any diagram D. If this were true, then Remark 7.6 would imply that the cone length of Tot M D is bounded above by the maximum of the following numbers:…”

“…On the other hand, the Khovanov spectrum of a link in its maximal extreme quantum grading is Spanier-Whitehead dual to that of its mirror image in minimal grading. This construction was exploded in [PS18] to make explicit computations of the Khovanov spectrum in (maximal) extreme quantum grading.…”

“…In [PS18] the second author conjectured (and proved for several cases) that the extreme Khovanov spectrum is always a wedge of spheres. If this conjecture were true, then the above decomposition would give an upper bound for the cone length of the spectrum X j almax D (Remark 7.8).…”

“…9 Following [PS18], given two vertices v and w in a graph, we say that v dominates w if the adjacent vertices to w are also adjacent to v.…”

Almost-extreme Khovanov spectra

“…Following[10], given two vertices v and w in a graph, we say that v dominates w if the adjacent vertices to w are also adjacent to v.…”