[Paper Review] Contact Surgeries on the Legendrian Figure-Eight Knot

James Conway|arXiv (Cornell University)|Oct 13, 2016

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper proves that all positive contact surgeries on any Legendrian figure-eight knot in the standard contact 3-sphere result in overtwisted contact structures. Using convex surface theory and invariants from Heegaard Floer homology—specifically the vanishing of the contact class—it shows that the induced contact structure on the knot complement is overtwisted, thereby implying the full surgery result is overtwisted.

ABSTRACT

We show that all positive contact surgeries on every Legendrian figure-eight knot in $(S^3, ξ_{ m{std}})$ result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.

Motivation & Objective

• To determine whether positive contact surgeries on Legendrian figure-eight knots in $(S^3, \xi_{\rm{std}})$ yield tight or overtwisted contact structures.
• To resolve a natural follow-up question to Lisca and Stipsicz's result on the vanishing of the Heegaard Floer contact class after $+1$ surgery.
• To establish a general method for detecting overtwistedness in contact surgeries using the contact class and convex surface techniques.
• To investigate the relationship between tightness and the non-vanishing of the Heegaard Floer contact class in positive contact surgeries.

Proposed method

• Apply convex surface theory to analyze the contact structure on $S^3 \setminus N(K)$, the knot complement, induced by positive contact surgery.
• Use the classification of tight contact structures on $S^3 \setminus N(K)$ with specific dividing curves on a convex Seifert surface.
• Construct two distinct tight contact structures on the complement: one with a bypass along the surface and one without, both shown to have non-vanishing Heegaard Floer contact class.
• Demonstrate that the contact structure induced by positive surgery on the figure-eight knot has a vanishing Heegaard Floer contact class.
• Leverage the fact that any tight contact structure on the complement must be contactomorphic to one with non-vanishing $EH$ invariant, leading to a contradiction if the surgery were tight.
• Use the amphichirality of the figure-eight knot and the vanishing of $\widehat{\mathcal{L}}(L)$ to show that the contact class of the surgery-induced structure vanishes.

Experimental results

Research questions

• RQ1 Does every positive contact surgery on a Legendrian figure-eight knot in $(S^3, \xi_{\rm{std}})$ produce an overtwisted contact structure?
• RQ2 Can the vanishing of the Heegaard Floer contact class be used to detect overtwistedness in positive contact surgeries?
• RQ3 Is there a general method to determine overtwistedness in positive contact surgeries using convex surface theory and contact invariants?
• RQ4 Does the non-vanishing of the Heegaard Floer contact class characterize tightness in positive contact surgeries on Legendrian knots in $(S^3, \xi_{\rm{std}})$?
• RQ5 Are the tight contact structures on the complement of the figure-eight knot classified, and can they be distinguished via their contact invariants?

Key findings

• All positive contact surgeries on any Legendrian figure-eight knot in $(S^3, \xi_{\rm{std}})$ result in overtwisted contact structures.
• The contact structure on the knot complement $S^3 \setminus N(K)$ induced by positive surgery has a vanishing Heegaard Floer contact class.
• There exist exactly two tight contact structures on $S^3 \setminus N(K)$ with the specified dividing curve configuration: one with a bypass along the surface and one without.
• Both of these tight contact structures on the complement have non-vanishing Heegaard Floer contact class, which contradicts the vanishing class of the surgery-induced structure.
• The contradiction implies that the surgery-induced structure on the complement must be overtwisted, hence the full surgery manifold is overtwisted.
• The result supports the conjecture that tightness of positive contact surgeries is equivalent to the non-vanishing of the Heegaard Floer contact class.