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2011 Planar open books with four binding components
Yankı Lekili
Algebr. Geom. Topol. 11(2): 909-928 (2011). DOI: 10.2140/agt.2011.11.909


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].


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Yankı Lekili. "Planar open books with four binding components." Algebr. Geom. Topol. 11 (2) 909 - 928, 2011.


Received: 17 September 2010; Accepted: 9 January 2011; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1220.57017
MathSciNet: MR2782547
Digital Object Identifier: 10.2140/agt.2011.11.909

Primary: 57R17

Keywords: binding number , contact structures , planar open books , right-veering

Rights: Copyright © 2011 Mathematical Sciences Publishers

Vol.11 • No. 2 • 2011
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