Volume 8, issue 2 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 Sr3(K) be the oriented 3–manifold obtained by rational r–surgery on a knot K S3. Using the contact Ozsváth–Szabó invariants we prove, for a class of knots K containing all the algebraic knots, that Sr3(K) carries positive, tight contact structures for every r2gs(K) 1, where gs(K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres Σ(2,3,4) and Σ(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m there exists a Seifert fibered rational homology 3–sphere Mm carrying at least m pairwise non–isomorphic tight, nonfillable contact structures.

Keywords
tight, fillable contact structures, Ozsváth–Szabó invariants
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57R57
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Publication
Received: 21 February 2004
Accepted: 29 May 2004
Published: 9 June 2004
Proposed: Peter Ozsváth
Seconded: John Morgan, Tomasz Mrowka
Authors
Paolo Lisca
Dipartimento di Matematica
Università di Pisa
I-56127 Pisa
Italy
András I Stipsicz
Rényi Institute of Mathematics
Hungarian Academy of Sciences
H-1053 Budapest
Reáltanoda utca 13–15
Hungary