[Paper Review] A transverse knot invariant from Z2-equivariant Heegaard Floer cohomology

Sungkyung Kang|arXiv (Cornell University)|Feb 1, 2018

Geometric and Algebraic Topology1 references3 citations

TL;DR

This paper introduces a new transverse link invariant, $ c_{\mathbb{Z}_2}(\xi_K) $, in $ \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K),p) $, the $ \mathbb{Z}_2 $-equivariant Heegaard Floer cohomology of the branched double cover of a transverse knot $ K \subset (S^3, \xi_{\text{std}}) $. The invariant is well-defined, functorial under symplectically constructible cobordisms, and detects nonvanishing of the non-equivariant contact class $ c(\xi_K) $, offering a new effective transverse invariant with potential to distinguish topologically isotopic, same-self-linking transverse knots.

ABSTRACT

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ along based knots, and then prove that our transverse link invariant $c_{\mathbb{Z}_{2}}(ξ_{K})$ is an well-defined element which is always nonvanishing and functorial under certain classes of symplectic cobordisms, and describe its behavior under negative stabilization. It follows that we can use properties of $c_{\mathbb{Z}_{2}}(ξ_{K})$ to give a condition on transverse knots K which implies the vanishing/nonvanishing of the contact class $c(ξ_{K})$.

Motivation & Objective

• To define a new transverse link invariant using $ \mathbb{Z}_2 $-equivariant Heegaard Floer cohomology.
• To establish naturality and functoriality of $ \widehat{HF}_{\mathbb{Z}_2} $ for branched double covers of based knots in $ S^3 $.
• To prove that the invariant $ c_{\mathbb{Z}_2}(\xi_K) $ is well-defined and functorial under symplectically constructible symplectic cobordisms.
• To relate the behavior of $ c_{\mathbb{Z}_2}(\xi_K) $ under negative stabilization to the nonvanishing of the standard contact class $ c(\xi_K) $.
• To investigate whether $ c_{\mathbb{Z}_2}(\xi_K) $ can distinguish topologically isotopic transverse knots with the same self-linking number.

Proposed method

• Construct the invariant $ c_{\mathbb{Z}_2}(\xi_K) $ as an element in $ \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K),p) $, the equivariant Heegaard Floer cohomology of the branched double cover of a transverse knot $ K $.
• Prove naturality of $ \widehat{HF}_{\mathbb{Z}_2} $ by showing it admits an action of the mapping class group $ MCG(S^3, K, p) $ for based knots.
• Define symplectically constructible symplectic cobordisms as compositions of weakly symplectically isotopic versions of isotopies, births, and saddles.
• Establish functoriality of $ c_{\mathbb{Z}_2}(\xi_K) $ by showing it is preserved under maps induced by such cobordisms.
• Use the behavior of $ c_{\mathbb{Z}_2}(\xi_K) $ under negative stabilization to derive a condition on the self-linking number implying nonvanishing of $ c(\xi_K) $.
• Analyze the structure of the set $ \{ c_{\mathbb{Z}_2}(\xi_T) \} $ for all transverse representatives $ T $ of a fixed knot type $ K $, showing it forms a $ \theta $-tower if the invariant is ineffective.

Experimental results

Research questions

• RQ1 Is $ c_{\mathbb{Z}_2}(\xi_K) $ an effective transverse invariant, capable of distinguishing topologically isotopic transverse knots with the same self-linking number?
• RQ2 Can $ c_{\mathbb{Z}_2}(\xi_K) $ be used to determine the nonvanishing of the standard contact class $ c(\xi_K) $?
• RQ3 Does the set of $ c_{\mathbb{Z}_2}(\xi_T) $ for all transverse representatives $ T $ of a knot $ K $ form a single $ \theta $-tower in $ \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K)) $?
• RQ4 Is there a non-constructible symplectic cobordism between two transverse knots in $ (S^3, \xi_{\text{std}}) $?
• RQ5 Can $ c^{-}(K) $ or $ \hat{c}(K) $ in knot Floer homology be recovered from $ c_{\mathbb{Z}_2}(\xi_K) $?

Key findings

• The invariant $ c_{\mathbb{Z}_2}(\xi_K) \in \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K),p) $ is well-defined and independent of the choice of basepoint, making it an invariant of the transverse isotopy class of $ K $.
• The $ \mathbb{F}_2[\theta] $-module $ \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K),p) $ is natural, admitting an action of $ MCG(S^3, K, p) $, and thus functorial under based cobordisms.
• For symplectically constructible symplectic cobordisms, the map $ \hat{f}_{(S,s)} $ preserves $ c_{\mathbb{Z}_2}(\xi_K) $, i.e., $ \hat{f}_{(S,s)}(c_{\mathbb{Z}_2}(\xi_{K_2})) = c_{\mathbb{Z}_2}(\xi_{K_1}) $.
• The invariant satisfies $ c_{\mathbb{Z}_2}(\xi_{K^{-}}) = \theta \cdot c_{\mathbb{Z}_2}(\xi_K) $ under negative stabilization, mirroring the behavior of $ c^{-}(K) $ in knot Floer homology.
• If $ c_{\mathbb{Z}_2}(\xi_K) $ is ineffective, then all transverse representatives of a knot $ K $ yield $ \theta $-towers in $ \widehat{HF}_{\mathbb{Z}_2}(\Sigma(K)) $, with the minimal-order element being a knot invariant.
• The existence of a non-constructible symplectic cobordism between transverse knots remains an open question, though the paper constructs a framework to investigate it.