Tight contact structures via admissible transverse surgery

Abstract

We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure [Formula: see text] is supported by the fibered knot [Formula: see text], we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure [Formula: see text] supported by the mirror knot [Formula: see text]. We derive several corollaries about the existence of tight contact structures, L-space knots outside [Formula: see text], nonplanar contact structures, and nonplanar Legendrian knots.