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