Tight Contact Structures via Admissible Transverse Surgery

TL;DR

This paper characterizes when negative surgeries on fibred transverse knots in contact 3-manifolds produce tight contact structures with non-vanishing Heegaard Floer invariants, linking properties of mirror knots and contact topology.

Contribution

It provides a new criterion for tightness after surgery based on contact invariants and mirror knot analysis, advancing understanding of contact structures in 3-manifolds.

Findings

Characterization of tight contact structures via Heegaard Floer invariants.

Corollaries on existence of tight structures and non-planar knots.

Insights into L-space knots outside S^3.

Abstract

We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $\xi_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure $\xi_{\overline{K}}$ supported by the mirror knot $\overline{K} \subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, non-planar contact structures, and non-planar Legendrian knots.