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 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 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.
Keywords
Cite
@article{arxiv.2111.05187,
title = {Braided open book decompositions in $S^3$},
author = {Benjamin Bode},
journal= {arXiv preprint arXiv:2111.05187},
year = {2023}
}
Comments
42 pages, 14 figures V2: There was an error in the proof of Theorem 1.5 of version 1 of this article. The corresponding section has been removed. Several other mistakes and inaccuracies have been fixed. The results (except Theorem 1.5 from version 1) have not been affected by these changes. V3: Some typos have been fixed, references and Figure 7 have been updated