Symplectic annular Khovanov homology and fixed point localizations

TL;DR

The authors define a new annular variant of symplectic Khovanov homology, AKh_symp, by introducing an annular filtration and proving its invariance under Markov moves. They establish spectral sequences from AKh_symp to Kh_symp and relate annular symplectic Kh to Heegaard Floer-type invariants in branched covers, including link Floer homology and two-periodic/strongly invertible knot contexts. The work develops two Z/2-quotients and axis-moving maps, yielding localization results and mapping-cone descriptions that generalize and unify several prior localization phenomena in both symplectic and combinatorial settings. Furthermore, the paper provides detailed constructions of axis-moving maps for strongly invertible knots, along with multiple illustrative examples (e.g., the trefoil and Hopf link) to demonstrate the framework’s computational aspects. The results collectively advance equivariant and annular versions of Khovanov-type theories in the symplectic category, with conjectural links to Mak–Smith and KLRW-type algebraic structures, and strengthen the bridge to Heegaard Floer theory via localization mechanisms.

Abstract

We introduce a new version of symplectic annular Khovanov homology and establish spectral sequences from (i) the symplectic annular Khovanov homology of a knot to the link Floer homology of the lift of the annular axis in the double branched cover; (ii) the symplectic Khovanov homology of a two-periodic knot to the symplectic annular Khovanov homology of its quotient; and (iii) the symplectic Khovanov homology of a strongly invertible knot to the cone of the axis-moving map between the symplectic annular Khovanov homology of the two resolutions of its quotient.

Figures (28)

• Figure 2.1: Left: A $2$-strand annular braid $\beta$, with strands drawn in black, and the axis $\{0\} \times \mathbb{R}$, drawn dashed and in green. Right: The closure $\operatorname{cl}(\beta)$ of $\beta$.
• Figure 2.2: The generator $\sigma_3$ in $\operatorname{Br}_{1,5}$
• Figure 2.3: Left: A diagram for the left-handed trefoil as a two-periodic knot. Right: The quotient knot (an unknot). The intersection of the axis of symmetry with the plane is marked with a star.
• Figure 2.4: Left: An intravergent diagram for the right-handed trefoil as a strongly invertible knot. Right: The quotient knot constructed using the choice of half-axis specified by the diagram.
• Figure 2.5: Resolutions and quotients of the intravergent diagram of Figure \ref{['fig:strongly_invertible_two']}. The top row shows the resolutions $K_0$ and $K_1$ on the left and right respectively. The bottom row shows their quotients $\overline{K}_0$ and $\overline{K}_1$. In all cases the intersection of the axis of symmetry, which is also the annular axis, with the plane is marked with a star.

Theorems & Definitions (117)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Theorem 2.1