Ozsváth–Szábo invariants and tight contact three-manifolds I

Lisca, Paolo¹ ; Stipsicz, András I²

¹ Dipartimento di Matematica, Università di Pisa, I-56127 Pisa, Italy
² Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda utca 13–15, Hungary

Geometry & topology, Tome 8 (2004) no. 2, pp. 925-945.

Résumé

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 $r\ne 2g_s\left(K\right)-1$, 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 $m\in \mathbb{N}$ there exists a Seifert fibered rational homology 3–sphere $M_m$ carrying at least $m$ pairwise non–isomorphic tight, nonfillable contact structures.

Détail

DOI : 10.2140/gt.2004.8.925

Keywords: tight, fillable contact structures, Ozsváth–Szabó invariants

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