Tight Contact Structures via Admissible Transverse Surgery

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.

arXiv comment17 pages, 3 figures; v2 has stronger results, more corollaries, shorter proofs