Algebraic & Geometric Topology

Tight contact structures and genus one fibered knots

John A Baldwin

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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]

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 701-735.

Received: 4 July 2006
Revised: 21 March 2007
Accepted: 21 March 2007
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57R17: Symplectic and contact topology 57R58: Floer homology
Secondary: 57R30: Foliations; geometric theory

contact structure Floer homology knot fibered taut foliation L-space


Baldwin, John A. Tight contact structures and genus one fibered knots. Algebr. Geom. Topol. 7 (2007), no. 2, 701--735. doi:10.2140/agt.2007.7.701.

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