For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L) \otimes_{ \mathbb{ F}_2} \mathbb{ F}_2[\theta, \theta^{ -1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{ \mathbb{ F}_2} \mathbb{ F}_2[\theta, \theta^{ -1}].$ This spectral sequence splits along quantum and $sl_2$ weight space gradings, proving a rank inequality $rank\ AKh^{ j,k}(L) \leq rank\ AKh^{ 2j-k,k} (\tilde L)$ 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.
Categories: math.GT math.QA