Admissible transverse surgery does not preserve tightness

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.

26 pages, 2 figures, references and discussion in the introduction and abstract corrected

References in corpus (5)

• Lectures on open book decompositions and contact structures
• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
• Heegaard Floer homologies and contact structures
• HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds