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

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

2004-04-06

arxiv.org/abs/math/0404135v2

Geom. Topol. 8(2004) 925-945

Mathematics

Symplectic Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper24.abs.html

Scientific paper

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.

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