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Tight contact structures and genus one fibered knots
John A Baldwin
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Algebraic & Geometric Topology 7
(2007) 701–735
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Abstract
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We study contact structures compatible with genus one open book decompositions
with one boundary component. Any monodromy for such an open book can be
written as a product of Dehn twists around dual nonseparating curves in the
once-punctured torus. Given such a product, we supply an algorithm to determine
whether the corresponding contact structure is tight or overtwisted for all but a small
family of reducible monodromies. We rely on Ozsváth–Szabó Heegaard Floer
homology in our construction and, in particular, we completely identify the
L–spaces with genus one, one boundary component, pseudo-Anosov open
book decompositions. Lastly, we reveal a new infinite family of hyperbolic
three-manifolds with no co-orientable taut foliations, extending the family
discovered by Roberts, Shareshian, and Stein in [J. Amer. Math. Soc. 16 (2003)
639–679]
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Keywords
contact structure, Floer homology, knot,
fibered, taut foliation, L-space
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Mathematical Subject Classification
Primary: 57M27, 57R17, 57R58
Secondary: 57R30
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Publication
Received: 4 July 2006
Revised: 21 March 2007
Accepted: 21 March 2007
Published: 30 May 2007
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