Extreme Khovanov spectra

Abstract

We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.

Almost-extreme Khovanov spectra

“…Shortly after, another geometric realization of the (minimal) extreme Khovanov homology was given [GMS18] in terms of an independence simplicial complex constructed from the link diagram. This construction was later shown to coincide with the one of Lipshitz and Sarkar [CMS20] as follows: functors from the cube to the Burnside 2-category can be understood as generalizations of simplicial complexes; such a functor is a simplicial complex if it factors through the category of sets and takes values on singletons and the empty set.…”

“…On the other hand, in [CMS20] the authors showed that if a cube F : 2 n → B factors through some functor F : Remark 3.6. The spectrum Tot F is denoted |F | in [LLS20].…”

“…Let S ⊂ Set be the full subcategory on ∅ and a singleton, and let S p ⊂ Set • be the full subcategory on the basepoint and S 0 . Downwards closed subposets of the cube are in bijection with functors 2 n → S. Subposets of the cube are in bijection with functors 2 n → S p (see discussion in the final section of [CMS20], for example). The realization of a subposet of the cube is the desuspension of the totalization of its associated functor 2 n → S p ⊂ Set • .…”

Geometric realization of the almost-extreme Khovanov homology of semiadequate links

“…Independently, in [GMS] it was introduced a method to associate to every link diagram D a simplicial complex I D so that its cohomology equals Khovanov homology of D at extreme quantum grading. This construction was proven to be equivalent to that of Lipshitz and Sarkar in extreme quantum grading [CS1]. The following conjecture was formulated in [PS1].…”

On bipartite circle graphs and Khovanov homology

“…Lipshitz and Sarkar constructed a Z-graded family of spectra X j (D)) associated to a link diagram D, and they proved that for each j ∈ Z, the stable homotopy type of the spectrum X j (D)) is a link invariant up to stable homotopy and there is a canonical isomorphism H * (X j (D)) ∼ = Kh * ,j (D) [5]. Morán and Silvero [6] later proved that X D is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading. Our proof uses combinatorial structures of a bipartite circle graph.…”