Khovanov homology of strongly invertible knots and their quotients (Q6630222)

Suppose that \( \rho : S^3 \rightarrow S^3 \) is a finite-order orientation-preserving diffeomorphism with fixed point set the (unknotted) circle \( C \hookrightarrow S^3 \). A knot \(K \hookrightarrow S^3 \) is \textit{strongly invertible} if \(K \cap C \) consists of two points and \( \rho \) preserves \( K \) setwise while reversing its orientation.\N\NStrongly invertible knots possess so-called \textit{intravergent} diagrams. Fix a marked point in the plane and consider rotation by \(\pi\) radians about this point; a diagram is intravergent if a crossing lies above the marked point and the diagram is preserved under the rotation.\N\NGiven an intravergent diagram \( D \) of a strongly invertible \(K \) let \( D_0 \), \( D_1 \) denote the diagrams obtained by resolving the crossing lying above the marked point into its \(0\)-\ or \(1\)-resolution, respectively. Denote by \( \overline{D_0} \), \( \overline{D_1} \) the quotients of these diagrams by the rotation (that preserves \(D\)). As \( \overline{D_0} \) and \( \overline{D_1} \) do not intersect the marked point one can consider them as lying on an annulus. Finally, denote by \( \overline{K_0} \), \( \overline{K_1} \) the annular knots represented by \( \overline{D_0} \), \( \overline{D_1} \), respectively.\N\NThe main result of this paper is the construction of a spectral sequence relating the Khovanov homology of \(K \) to the annular Khovanov homology of \( \overline{K_0} \) and \( \overline{K_1} \). Specifically, the spectral sequence abuts to the mapping cone over \( f^{+} : ACKh ( \overline{K_1} ) \rightarrow ACKh ( \overline{K_0} ) \), where \( f^{+} \) is a certain anchored cobordism as introduced by \textit{R. Akhmechet} and \textit{M. Khovanov} [Algebr. Geom. Topol. 23, No. 7, 3129--3204 (2023; Zbl 07786959)].\N\NVarious rank inequalities are deduced using this spectral sequence, which in turn are used to distinguish slice discs. The main result is proved using techniques similar to work of Stoffregen-Zhang and Borodzik-Politarczyk-Silvero in the periodic case [\textit{M. Stoffregen} and \textit{M. Zhang}, Geom. Topol. 28, No. 4, 1501--1585 (2024; Zbl 07904156); \textit{M. Borodzik} et al., Math. Ann. 380, No. 3--4, 1233--1309 (2021; Zbl 1506.57006)].\N\NFor the entire collection see [Zbl 1545.53005].