Planar open books with four binding components
arXiv:1008.3529 · doi:10.2140/agt.2011.11.909
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 S^3 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 Ozsvath-Szabo 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, Matic from arXiv:0609734 .
19 pages, 10 figures. Two cancelling sign errors removed
References in corpus (9)
- Géométrie de contact: de la dimension trois vers les dimensions supérieures
- Right-veering diffeomorphisms of compact surfaces with boundary II
- Lectures on open book decompositions and contact structures
- Weak Symplectic Fillings and Holomorphic Curves
- Planar open book decompositions and contact structures
- Inhomogeneities on the surface of 21 Lutetia, the asteroid target of the Rosetta mission
- Planar open books and Floer homology
- Mapping class group relations, Stein fillings, and planar open book decompositions
- A note on the support norm of a contact structure