Ozsváth–Szábo invariants and tight contact three-manifolds I

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 $r\ne 2g_s\left(K\right)-1$, 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.