Symplectic annular Khovanov homology and fixed point localizations

TL;DR

This paper develops a new symplectic annular Khovanov homology theory and constructs spectral sequences linking it to various other knot invariants, enhancing understanding of knot symmetries and fixed point phenomena.

Contribution

It introduces a novel version of symplectic annular Khovanov homology and establishes spectral sequences connecting it to link Floer homology and symplectic Khovanov homology for symmetric knots.

Findings

Spectral sequence from symplectic annular Khovanov homology to link Floer homology.

Spectral sequence relating symplectic Khovanov homology of periodic knots to annular homology.

Construction of a cone of the axis-moving map for strongly invertible knots.

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.