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

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}].$Thisspectralsequencesplitsalongquantumand$sl_2$ weight space gradings, proving a rank inequality $rank\ AKh^{j,k}(L) \leq rank
AKh^{2j-k,k} (\tilde L)$foreverypairofquantumand$sl_2$ 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.