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For a 2-periodic link $\tilde L$ in the thickened annulus and its quotientlink $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L)\otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta, \theta^{-1}] \rightrightarrowsE^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta,\theta^{-1}].$ This spectral sequence splits along quantum and $sl_2$ weightspace 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 spacegradings $(j,k)$. We also present a few decategorified consequences and discusspartial results toward a similar statement for the Khovanov homology of2-periodic links.

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