Admissible transverse surgery does not preserve tightness

TL;DR

This paper demonstrates that admissible transverse surgery can destroy tightness in contact 3-manifolds, introduces a new invariant for transverse knots, and explores complex phenomena in contact topology.

Contribution

It provides the first examples of tight contact manifolds losing tightness after admissible transverse surgery and clarifies its relationship with Legendrian surgery.

Findings

Admissible transverse surgery can produce overtwisted manifolds from tight ones.

Introduces a new invariant based on slopes for transverse knots.

Identifies hyperbolic, universally tight manifolds with vanishing Heegaard Floer invariants.

Abstract

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots - namely, the range of slopes on which admissible transverse surgery preserves tightness - and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universally tight contact 3-manifolds whose Heegaard Floer contact invariants vanish (and which are not weakly fillable); and the existence of open books with arbitrarily high fractional Dehn twist coefficients whose compatible contact structures are not deformations of co-orientable taut foliations.