Localization and the link Floer homology of doubly-periodic knots

TL;DR

This paper develops equivariant Heegaard diagrams for q-periodic knots, derives spectral sequences relating their link Floer homologies, and applies these to classical bounds on knot genus and fibredness.

Contribution

It introduces new spectral sequences connecting the Floer homologies of periodic knots and their quotients, providing novel tools for knot genus and fibredness analysis.

Findings

Spectral sequences relate the Floer homologies of periodic knots and their quotients.

Classical bounds on knot genus are recovered using these spectral sequences.

A weak version of fibredness results is established for doubly-periodic knots.

Abstract

A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The quotient of \widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For \widetilde{K} a two-periodic knot, we show there is a spectral sequence whose E^1 page is \hat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes \mathbb Z_2((\theta)) and whose E^{\infty} pages is isomorphic to (\hat{\mathit{HFL}}(S^3,K\cup U)\otimes V^{\otimes (n-1)})\otimes \mathbb Z_2((\theta)), as \mathbb Z_2((\theta))-modules, and a related spectral sequence whose E^1 page is (\hat{\mathit{HFK}}(S^3,\widetilde{K})\otimes V^{\otimes (2n-1)}\otimes W)\otimes \mathbb Z_2((\theta)) and whose E^{\infty} page is isomorphic to (\hat{\mathit{HFK}}(S^3,K)\otimes V^{\otimes (n-1)} \otimes W)\otimes \mathbb Z_2((\theta)). As a consequence, we use these spectral sequences to recover a classical lower bound of Edmonds on the genus of \widetilde{K}, along with a weak version of a classical fibredness result of Edmonds and Livingston.