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 $L$ –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]

Keywords

contact structure, Floer homology, knot, fibered, taut foliation, L-space

Mathematical Subject Classification 2000

Primary: 57M27, 57R17, 57R58

Secondary: 57R30

References

Publication

Received: 4 July 2006

Revised: 21 March 2007

Accepted: 21 March 2007

Published: 30 May 2007

Authors

John A Baldwin
Department of Mathematics Columbia University New York, NY 10027
http://www.math.columbia.edu/~baldwin

Volume 7, issue 2 (2007)

John A Baldwin

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors