Algebraic & Geometric Topology

Planar open books with four binding components

Yankı Lekili

Full-text: Open access

Abstract

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].

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 2 (2011), 909-928.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715213

Digital Object Identifier
doi:10.2140/agt.2011.11.909

Mathematical Reviews number (MathSciNet)
MR2782547

Zentralblatt MATH identifier
1220.57017

Subjects
Primary: 57R17: Symplectic and contact topology

Keywords
planar open books contact structures right-veering binding number

Citation

Lekili, Yankı. Planar open books with four binding components. Algebr. Geom. Topol. 11 (2011), no. 2, 909--928. doi:10.2140/agt.2011.11.909. https://projecteuclid.org/euclid.agt/1513715213


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