Volume 7, issue 2 (2007)

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Tight contact structures and genus one fibered knots

John A Baldwin

Algebraic & Geometric Topology 7 (2007) 701–735

arXiv: math.SG/0604580


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]

contact structure, Floer homology, knot, fibered, taut foliation, L-space
Mathematical Subject Classification 2000
Primary: 57M27, 57R17, 57R58
Secondary: 57R30
Received: 4 July 2006
Revised: 21 March 2007
Accepted: 21 March 2007
Published: 30 May 2007
John A Baldwin
Department of Mathematics
Columbia University
New York, NY 10027