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.

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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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