Abstract
We introduce a functor from the cube to the Burnside 2-category and prove that it is equivalent to the Khovanov spectrum given by Lipshitz and Sarkar in the almost-extreme quantum grading. We provide a decomposition of this functor into simplicial complexes. This decomposition allows us to compute the homotopy type of the almost-extreme Khovanov spectra of diagrams without alternating pairs.
Highlights
Khovanov homology is a powerful link invariant introduced by Mikhail Khovanov in [4] as a categorification of the Jones polynomial
[7] Lipshitz and Sarkar refined this invariant to obtain, for each quantum grading j, a spectrum XDj whose stable homotopy type is a link invariant and whose cohomology is isomorphic to the Khovanov homology of the link, i.e., H ∗(XDj ) ∼= K h∗, j (L)
In 2018 the second author, together with Przytycki applied these techniques one step further, to the almost-extreme quantum grading. They restricted to the study of 1-adequate link diagrams, i.e., those whose 1-resolution contains no chords with both endpoints in the same circle, and built a pointed semi-simplicial set whose homology is isomorphic to the almost-extreme Khovanov homology of the diagram: Theorem ([11]) If D is a 1-adequate link diagram, (i) The functor FDjalmax gives rise to a pointed semi-simplicial set. (ii) The realization of FDjalmax has the homotopy type of a wedge of spheres and possibly a copy of asuspension of RP2
Summary
Khovanov homology is a powerful link invariant introduced by Mikhail Khovanov in [4] as a categorification of the Jones polynomial. Given an oriented diagram D representing a link L, he constructed a finite Z-graded family of chain complexes. Whose bigraded homology groups, K hi, j (D), are link invariants. The groups K hi, j (L) are known as Khovanov homology groups of L, and the indexes i and j as homological and quantum gradings, respectively.
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