#### Geometry & Topology, Vol. 8 (2004)
Paper no. 24, pages 925--945.

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

### Paolo Lisca Andras I Stipsicz

**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.
**Keywords**.
Tight, fillable contact structures, Ozsvath-Szabo invariants

**AMS subject classification**.
Primary: 57R17.
Secondary: 57R57.

**DOI:** 10.2140/gt.2004.8.925

**E-print:** `arXiv:math.SG/0404135`

Submitted to GT on 21 February 2004.
Paper accepted 29 May 2004.
Paper published 9 June 2004.

Notes on file formats
Paolo Lisca Andras I Stipsicz

Dipartimento di Matematica, Universita di Pisa

I-56127 Pisa, ITALY

and

Renyi Institute of Mathematics, Hungarian Academy of Sciences

H-1053 Budapest, Realtanoda utca 13--15, Hungary

Email: lisca@dm.unipi.it, stipsicz@math-inst.hu

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