Ozsvath-Szabo invariants and tight contact three-manifolds, I

Abstract

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Sigma(2,3,4) and -Sigma(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m in N there exists a Seifert fibered rational homology 3-sphere M_m carrying at least m pairwise non-isomorphic tight, nonfillable contact structures.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper24.abs.html

References in corpus (13)

• Knot Floer homology and the four-ball genus
• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Holomorphic disks and genus bounds
• Lectures on open book decompositions and contact structures
• Heegaard Floer homologies and contact structures
• Transverse contact structures on Seifert 3-manifolds
• Surgery diagrams for contact 3-manifolds
• Seifert fibered contact three-manifolds via surgery
• Planar open books and Floer homology
• An infinite family of tight, not semi-fillable contact three-manifolds
• The Weinstein Conjecture for Planar Contact Structures in Dimension Three
• On Plumbed L-spaces
• Classification of tight contact structures on small Seifert 3-manifolds with e_0\geq 0

Cited by in corpus (27)

• Khovanov homology detects the trefoils
• On the existence of tight contact structures on Seifert fibered 3-manifolds
• The vanishing of the contact invariant in the presence of torsion
• Handle moves in contact surgery diagrams
• Distinguishing topologically and smoothly doubly slice knots
• A combinatorial description of the Heegaard Floer contact invariant
• Monopole Floer homology and Legendrian knots
• Stein fillable Seifert fibered 3-manifolds
• Dehn Twists in Heegaard Floer Homology
• Contact surgery and supporting open books
• Non-looseness of non-loose knots
• Nuclei and exotic 4-manifolds
• Infinitely many universally tight contact manifolds with trivial Ozsvath-Szabo contact invariants
• Ozsváth-Szabó invariants of contact surgeries
• Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot
• Combinatorial Reeb dynamics on punctured contact 3-manifolds
• Hyperbolic 3-manifolds admitting no fillable contact structures
• Tight Planar Contact Manifolds with Vanishing Heegaard Floer Contact Invariants
• Symmetries and adjunction inequalities for knot Floer homology
• Open books and plumbings
• Contact Ozsvath-Szabo Invariants and Giroux Torsion
• Contact surgery graphs
• Ozsvath-Szabo invariants and fillability of contact structures
• Contact (+1)-surgeries on rational homology 3-spheres
• On naturality of the Ozsvath-Szabo contact invariant
• Rational cuspidal curves and symplectic fillings
• L-spaces, taut foliations and fibered hyperbolic two-bridge links