[Paper Review] A note on the link Floer homology of doubly-periodic knots

Kristen Hendricks|arXiv (Cornell University)|Jun 26, 2012

Geometric and Algebraic Topology25 references3 citations

TL;DR

This paper constructs equivariant Heegaard diagrams for q-periodic knots, using them to derive Murasugi's condition on Alexander polynomials and establish spectral sequences relating the equivariant link Floer homology of the periodic knot $;tilde{K} \cup U$ to that of the quotient knot $K \cup U$. The key result is a spectral sequence that recovers Edmonds' genus lower bound and a weak fibredness criterion for 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(( heta)) 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(( heta)), as \mathbb Z_2(( heta))-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(( heta)) 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(( heta)). 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.

Motivation & Objective

• To develop a framework for studying periodic knots using equivariant Heegaard diagrams.
• To re-derive Murasugi's classical condition on Alexander polynomials of periodic knots using this new diagrammatic approach.
• To construct spectral sequences linking the equivariant link Floer homology of the total space to that of the quotient knot.
• To recover topological invariants such as genus lower bounds and fibredness criteria from these spectral sequences.
• To extend the understanding of periodic knot invariants through link Floer homology and equivariant structures.

Proposed method

• Construct equivariant Heegaard diagrams for q-periodic knots that respect the $Δ_q$-action.
• Use the diagrammatic structure to derive the Alexander polynomial condition via equivariant grading shifts.
• Define a spectral sequence whose $E^1$ page is $\widehat{\mathit{HFL}}(S^3,\widetilde{K}\cup U) \otimes V^{\otimes (2n-1)} \otimes \mathbb{Z}_2((\theta))$.
• Show that the $E^\infty$ page is isomorphic to $(\widehat{\mathit{HFL}}(S^3,K\cup U) \otimes V^{\otimes (n-1)}) \otimes \mathbb{Z}_2((\theta))$ as $\mathbb{Z}_2((\theta))$-modules.
• Construct a related spectral sequence for knot Floer homology with $E^1$ page $\widehat{\mathit{HFK}}(S^3,\widetilde{K}) \otimes V^{\otimes (2n-1)} \otimes W \otimes \mathbb{Z}_2((\theta))$ and $E^\infty$ page $\widehat{\mathit{HFK}}(S^3,K) \otimes V^{\otimes (n-1)} \otimes W \otimes \mathbb{Z}_2((\theta))$.
• Use the spectral sequences to deduce topological constraints on the periodic knot, including genus bounds and fibredness conditions.

Experimental results

Research questions

• RQ1 How can equivariant Heegaard diagrams be constructed for q-periodic knots to reflect the $Δ_q$-action?
• RQ2 Can Murasugi's condition on the Alexander polynomial of periodic knots be derived from these diagrams?
• RQ3 What spectral sequences relate the equivariant link Floer homology of the total space to that of the quotient knot?
• RQ4 How do these spectral sequences recover classical invariants like Edmonds' genus lower bound?
• RQ5 Can a weak fibredness result for periodic knots be deduced from the spectral sequence structure?

Key findings

• The spectral sequence from $\widehat{\mathit{HFL}}(S^3,\widetilde{K}\cup U) \otimes V^{\otimes (2n-1)} \otimes \mathbb{Z}_2((\theta))$ to $(\widehat{\mathit{HFL}}(S^3,K\cup U) \otimes V^{\otimes (n-1)}) \otimes \mathbb{Z}_2((\theta))$ establishes a direct link between equivariant and quotient link Floer homology.
• The spectral sequence for knot Floer homology has $E^1$ page $\widehat{\mathit{HFK}}(S^3,\widetilde{K}) \otimes V^{\otimes (2n-1)} \otimes W \otimes \mathbb{Z}_2((\theta))$ and $E^\infty$ page $\widehat{\mathit{HFK}}(S^3,K) \otimes V^{\otimes (n-1)} \otimes W \otimes \mathbb{Z}_2((\theta))$, preserving the module structure.
• The existence of these spectral sequences implies that the rank of $\widehat{\mathit{HFK}}(S^3,K)$ is bounded above by the rank of $\widehat{\mathit{HFK}}(S^3,\widetilde{K})$, reflecting the quotient structure.
• Edmonds' classical lower bound on the genus of $\widetilde{K}$ is recovered as a consequence of the spectral sequence's convergence and grading structure.
• A weak version of the fibredness result of Edmonds and Livingston for periodic knots is deduced from the spectral sequence's behavior on the $\mathbb{Z}_2((\theta))$-module structure.
• The construction provides a new diagrammatic and homological proof of Murasugi's condition on the Alexander polynomial of periodic knots, using equivariant Heegaard diagrams.