[Paper Review] $\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomology of knots in $S^{3}$ as a strong Heegaard invariant

Sungkyung Kang|arXiv (Cornell University)|Oct 3, 2018

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper establishes that $̂{HF}_{\mathbb{Z}_2}(\Sigma(K))$, the $\mathbb{Z}_2$-equivariant Heegaard Floer cohomology of the branched double cover of a knot $K \subset S^3$, is a strong Heegaard invariant by showing it can be computed from knot Heegaard diagrams and extended bridge diagrams of arbitrary genus. The key contribution is the construction of a transverse knot invariant $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ in $\widehat{HFK}_{\mathbb{Z}_2}(\Sigma(K),K)$, which refines both the LOSS invariant and the $\mathbb{Z}_2$-equivariant contact class.

ABSTRACT

The $\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomlogy $\widehat{HF}_{\mathbb{Z}_{2}}(Σ(K))$ of a knot $K$ in $S^{3}$, constructed by Hendricks, Lipshitz, and Sarkar, is an isotopy invariant which is defined using bridge diagrams of $K$ drawn on a sphere. We prove that $\widehat{HF}_{\mathbb{Z}_{2}}(Σ(K))$ can be computed from knot Heegaard diagrams of $K$ and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant $\hat{\mathcal{T}}_{\mathbb{Z}_{2}}(K)$ as an element of $\widehat{HFK}_{\mathbb{Z}_{2}}(Σ(K),K)$, which is a refinement of $\widehat{HF}_{\mathbb{Z}_{2}}(Σ(K))$, and show that it is a refinement of both the LOSS invariant $\hat{\mathcal{T}}(K)$ and the $\mathbb{Z}_{2}$-equivariant contact class $c_{\mathbb{Z}_{2}}(ξ_{K})$.

Motivation & Objective

• To establish that $\widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ is a strong Heegaard invariant, not just a weak one, by showing its invariance under all standard Heegaard moves.
• To demonstrate that $\widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ can be computed from knot Heegaard diagrams and extended bridge diagrams of arbitrary genus, not just sphere-based bridge diagrams.
• To construct a new transverse knot invariant $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ in $\widehat{HFK}_{\mathbb{Z}_2}(\Sigma(K),K)$ that refines existing invariants.
• To prove that the new invariant $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ is a refinement of both the LOSS invariant $\hat{\mathcal{T}}(K)$ and the $\mathbb{Z}_2$-equivariant contact class $c_{\mathbb{Z}_2}(\xi_K)$.

Proposed method

• The paper uses extended bridge diagrams on surfaces of arbitrary genus to represent knots in $S^3$, generalizing the original sphere-based bridge diagram construction.
• It constructs the $\mathbb{Z}_2$-equivariant Heegaard Floer cohomology $\widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ from these diagrams, leveraging the $\mathbb{Z}_2$-action induced by the branched double cover.
• The authors prove that the invariant is independent of the choice of diagram by showing invariance under isotopies and arc-slides (including types I–IV) in extended arc diagrams.
• They define a canonical element $EH_{\mathbb{Z}_2}$ in the equivariant chain complex using the Honda-Kazez-Mark construction on extended arc diagrams.
• The transverse invariant $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ is defined as the cohomology class of a specific chain-level representative derived from a multi-pointed open book structure.
• The naturality and functoriality of the construction are established via a natural isomorphism between the bridge-based and strong Heegaard invariant formulations.

Experimental results

Research questions

• RQ1 Can $\widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ be computed from knot Heegaard diagrams of arbitrary genus, not just sphere-based bridge diagrams?
• RQ2 Is $\widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ a strong Heegaard invariant, satisfying the full set of commutativity axioms under Heegaard moves?
• RQ3 Can a transverse knot invariant be constructed in $\widehat{HFK}_{\mathbb{Z}_2}(\Sigma(K),K)$ that refines both the LOSS invariant and the $\mathbb{Z}_2$-equivariant contact class?
• RQ4 Is the new invariant $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ invariant under transverse isotopy and compatible with the standard contact structure on $S^3$?
• RQ5 Does the natural map $\widehat{HFK}_{\mathbb{Z}_2}(\Sigma(K),K) \to \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ send $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ to $c_{\mathbb{Z}_2}(\xi_K)$?

Key findings

• Theorem 1.1 establishes that $\widehat{HF}_{\mathbb{Z}_2}(\mathcal{E}) \simeq \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ for any weakly admissible extended bridge diagram $\mathcal{E}$ with at least two A-arcs, generalizing the original construction.
• Theorem 1.2 proves the existence of an invertible natural transformation between the bridge-based and strong Heegaard invariant formulations of $\widehat{HF}_{\mathbb{Z}_2}$, confirming its naturality and strong invariance.
• Theorem 1.3 extends the result to all weakly admissible extended bridge diagrams, showing that $\widehat{HF}_{\mathbb{Z}_2}(\mathcal{E}) \simeq \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K))$ universally.
• The construction of $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ as a cohomology class in $\widehat{HFK}_{\mathbb{Z}_2}(\Sigma(K),K)$ is shown to be invariant under transverse isotopy of $K$.
• Theorem 6.15 confirms that $\hat{\mathcal{T}}_{\mathbb{Z}_2}(K)$ maps to $c_{\mathbb{Z}_2}(\xi_K)$ under the natural projection, proving it is a refinement of the $\mathbb{Z}_2$-equivariant contact class.
• The paper establishes a localization isomorphism for $\widehat{HFK}_{\mathbb{Z}_2}$, extending the formalism to the knot Floer setting.