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In this note, we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book, we given an application of this result to show that there are planar open books of the standard contact structure on  S  3  with five (or any number larger than five) boundary components that do not destabilize. We also show that any planar open book with four or fewer boundary components does destabilize.

The Arc Complex and Contact Geometry: Nondestabilizable Planar Open Book Decompositions of the Tight Contact 3-Sphere