Volume 11, issue 2 (2011)

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Planar open books with four binding components

Yankı Lekili

Algebraic & Geometric Topology 11 (2011) 909–928

We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of S3 supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsváth–Szabó contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez and Matić from [J. Differential Geom. 83 (2009) 289–311].

planar open books, contact structures, right-veering, binding number
Mathematical Subject Classification 2010
Primary: 57R17
Received: 17 September 2010
Accepted: 9 January 2011
Published: 25 March 2011
Yankı Lekili
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road