Braided open book decompositions in $S^3$

TL;DR

This paper proves the equivalence of four different notions of braiding in open book decompositions of the 3-sphere and shows that all such open books with low braid index can be braided.

Contribution

It establishes the equivalence of four different definitions of braided open books in $S^3$ and characterizes those with braid index at most 3.

Findings

Four notions of braiding are equivalent in $S^3$.

All open books with binding braid index ≤ 3 are braided.

Unified framework for understanding braided open books in $S^3$.

Abstract

We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be defined to 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.