Braided open book decompositions in $S^3$

Abstract

We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be braided. These include generalised exchangeability defined by Morton and Rampichini and mutual braiding defined by Rudolph, which were shown to be equivalent by Rampichini, as well as P-fiberedness and a property related to simple branched covers of $S^3$ inspired by work of Montesinos and Morton. We prove that these four notions of a braided open book are actually all equivalent to each other. We show that all open books in the 3-sphere whose binding has a braid index of at most 3 can be braided in this sense. Furthermore, every open book whose page can be obtained from a disk via a sequence of Hopf plumbings without any deplumbings can be braided.