Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 7, Number 2 (2007), 701-735.
Tight contact structures and genus one fibered knots
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 –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]
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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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