A rank inequality for the annular Khovanov homology of 2–periodic links

Abstract

For a 2-periodic linkL in the thickened annulus and its quotient link L, we exhibit a spectral sequence with. This spectral sequence splits along quantum and sl 2 weight space gradings, proving a rank inequality rk F2 AKh j,k (L) ≤ rk F2 AKh 2j−k,k (L) for every pair of quantum and sl 2 weight space gradings (j, k). We also present a few decategorified consequences and discuss partial results toward a similar statement for the Khovanov homology of 2-periodic links.

Khovanov homotopy type, periodic links and localizations

“…The special case (p = 2) of Corollary 1.5 was proved by Zhang [47]. She also proved Corollary 1.6 for p = 2 and certain class of periodic links.…”

Categorical lifting of the Jones polynomial: a survey

“…There are many other spectral sequences from Khovanov homology, including more variants of the Lee spectral sequence [41,13], spectral sequences defined using instanton and monopole Floer homology [38,156,24], other spectral sequences defined via variants of Heegaard Floer homology [58,149], spectral sequences coming from equivariant symplectic Khovanov homology and equivariant Khovanov homology [162,35,185,168], and a combinatorial spectral sequence conjectured to agree with the Ozsváth-Szabó spectral sequence [170] (see also [155]). This last spectral sequence also supports another conjecture: that the Ozsváth-Szabó spectral sequence preserves the δ-grading δ = j − 2i on Khovanov homology [56].…”

Khovanov homology of strongly invertible knots and their quotients

“…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].…”