Volume 8, issue 2 (2004)

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Ozsváth–Szábo invariants and tight contact three-manifolds I

Paolo Lisca and András I Stipsicz

Geometry & Topology 8 (2004) 925–945
 arXiv: math.SG/0404135
Abstract

Let ${S}_{r}^{3}\left(K\right)$ be the oriented 3–manifold obtained by rational $r$–surgery on a knot $K\subset {S}^{3}$. Using the contact Ozsváth–Szabó invariants we prove, for a class of knots $K$ containing all the algebraic knots, that ${S}_{r}^{3}\left(K\right)$ carries positive, tight contact structures for every $r\ne 2{g}_{s}\left(K\right)-1$, where ${g}_{s}\left(K\right)$ is the slice genus of $K$. This implies, in particular, that the Brieskorn spheres $-\Sigma \left(2,3,4\right)$ and $-\Sigma \left(2,3,3\right)$ 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.

Keywords
tight, fillable contact structures, Ozsváth–Szabó invariants
Primary: 57R17
Secondary: 57R57