## Algebraic & Geometric Topology

### Tight contact structures and genus one fibered knots

John A Baldwin

#### Abstract

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

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

Dates
Revised: 21 March 2007
Accepted: 21 March 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796702

Digital Object Identifier
doi:10.2140/agt.2007.7.701

Mathematical Reviews number (MathSciNet)
MR2308961

Zentralblatt MATH identifier
1202.57013

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796702

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