Localization and the Floer homology of strongly invertible knots

TL;DR

This paper introduces spectral sequences in knot Floer homology for strongly invertible knots, leading to new invariants and insights into their topological properties.

Contribution

It establishes two spectral sequences related to strongly invertible knots and defines a new numerical invariant based on these sequences.

Findings

Spectral sequences connect knot Floer homology to simpler invariants.

A new numerical invariant for strongly invertible knots is introduced.

The methods provide tools for analyzing knot symmetries and invariants.

Abstract

We establish two spectral sequences in knot Floer homology associated to a directed strongly invertible knot K: one from the knot Floer homology of K to a two dimensional vector space, and one from the singular knot Floer homology of a singular knot associated to K to the knot Floer homology quotient knot of K. The first of these spectral sequences is used to define a numerical invariant of strongly invertible knots.