Khovanov homotopy type, periodic links and localizations

Abstract

Given an m-periodic link L ⊂ S 3 , we show that the Khovanov spectrum XL constructed by Lipshitz and Sarkar admits a homology group action. We relate the Borel cohomology of XL to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of XL 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 Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.Define X Kh (D) = q X q Kh (D). We will often write X Kh (L) instead of X Kh (D) noting that X Kh (L) is defined up to stable homotopy. The space X Kh (L) is called a Khovanov homotopy

Localization in Khovanov homology

“…Note also that in a recent preprint, Borodzik-Politarczyk-Silvero [BPS18] use equivariant flow categories to also show that X 0 ( L) = X e ( L) admits a Z p -action; the main Theorem 1.2 of [BPS18] is the first sentence of Theorem 1.3 in the present paper, although it is not clear that the action constructed in [BPS18] and that constructed here (in the case n = 0) agree. In [BPS18], they further relate the Borel equivariant cohomology of X e ( L) to Politarczyk's equivariant Khovanov homology [Pol17]. Jeff Musyt has also constructed a Z p -equivariant Khovanov stable homotopy type using methods similar to ours.…”

“…This would be useful to understand in order to relate the constructions of this paper with the equivariant Khovanov homology (or an odd version of same) constructed by Politarczyk [Pol17]. In more generality, it would be desirable to better understand a Z p -equivariant cell decomposition of X n ( L), so that, for example, the space X 0 ( L) could be related to the space constructed in [BPS18]. (q-2) A better understanding of the case of even p for the odd Khovanov-Burnside functor KHO would be desirable.…”

“…(q-3) Are there applications of the construction of the present paper to showing that some links are not periodic? Borodzik-Politarczyk-Silvero [BPS18] have obtained such applications; are there further applications that require using the odd theory? (q-4) Willis [Wil17] showed that the Khovanov homotopy type of torus links T (n, m) stabilizes as m → ∞.…”

Categorical lifting of the Jones polynomial: a survey

“…The proof of the first statement uses Kronheimer-Mrowka's spectral sequence, the second uses Dowlin's spectral sequence, and the third uses an annular version of the Kronheimer-Mrowka spectral sequence [179,181], Dowlin's spectral sequence, the spectral sequences for periodic knots [168,25] mentioned above, further hard results on Floer homology [179,181,97,104,36], and the sl 2 (C)-action on annular Khovanov homology [57]. Some of these, like the spectral sequences from equivariant Khovanov homology, lift, or at least recall, well-known properties of the Jones polynomial, such as K. Murasugi's formula [126].…”

“…There is an analogue for odd Khovanov homology [154] (see also [144]). The homotopical refinement can even be used to prove new results about Khovanov homology itself, such as formulas relating the Khovanov homology of periodic links and their quotients [168,25], partially lifting results of Murasugi [126].…”