  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 .
