Khovanov homotopy type, periodic links and localizations

Abstract

Given an m-periodic link $$L\subset S^3$$ L ⊂ S 3 , we show that the Khovanov spectrum $$\mathcal {X}_L$$ X L constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of $$\mathcal {X}_L$$ X L to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of $$\mathcal {X}_L$$ X L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.

Khovanov homology of strongly invertible knots and their quotients

“…The main work is to compute the E ∞ -page of the spectral sequence (Item (Θ-3)); the other properties are immediate. The strategy is similar to Stoffregen-Zhang's [SZ] and Borodzik-Politarczyk-Silvero's [BPS21]: we prove that the fixed points of the Z/2-action induced by the strong inversion on a CW complex representing the Khovanov stable homotopy type of K is related to the annular Khovanov stable homotopy type of (K 1 , K 0 ), and then apply classical Smith theory.…”

“…Similar results have been proved before, for other kinds of symmetries. Stoffregen-Zhang [SZ] and Borodzik-Politarczyk-Silvero [BPS21] showed that there is a spectral sequence relating the Khovanov homology of a periodic knot (a knot preserved by rotation around an axis disjoint from it) and the annular Khovanov homology of its quotient (see also [Cor,Zha18]). An analogous result relating the symplectic Khovanov homology of a 2-periodic knot and of its quotient was proved earlier by Seidel-Smith [SS10].…”

Categorical lifting of the Jones polynomial: a survey

“…The proof of the first statement uses Kronheimer and Mrowka's spectral sequence, the second uses Dowlin's spectral sequence, and the third uses an annular version of Kronheimer and Mrowka's spectral sequence [179,181], Dowlin's spectral sequence, the spectral sequences for periodic knots [26,167] mentioned above, further hard results on Floer homology [37,97,102,179,181] and the sl 2 (C)-action on annular Khovanov homology [57].…”

Stable homotopy refinement of quantum annular homology

“…We emphasize some differences and similarities between this paper and others appearing in the literature. There is no group action on the links considered in this paper, so our box maps and homotopy coherent refinements are different from those in [6,21,24]. Functors 2 n → B Z/2Z are considered in [23], and there the authors introduce actions of Z/2Z and Z/2Z × Z/2Z, which act internally on each box.…”

“…It is important to note that no group action is assumed to be present on the link L ⊂ A×I, so the context for our work is different from that in [6,21,24]. Therefore X r Aq (L) may be thought of as an "equivariant refinement" of X A (L), a structure that is not apparent in other constructions of the annular spectrum X A (L).…”