Homogeneous braids are visually prime

TL;DR

This paper proves that the closures of homogeneous braids are visually prime and introduces a new criterion for the primeness of open books, with implications for 3-manifold topology.

Contribution

It establishes that homogeneous braid closures are visually prime and develops a novel criterion for open book primeness, exploring its preservation under various operations.

Findings

Homogeneous braid closures are visually prime.

Primeness is not always preserved under Murasugi sums or stabilizations.

Figure-eight knot plumbings preserve primeness.

Abstract

We show that closures of homogeneous braids are visually prime, addressing a question of Cromwell. The key technical tool for the proof is the following criterion concerning primeness of open books, which we consider to be of independent interest. For open books of 3-manifolds the property of having no fixed essential arcs is preserved under essential Murasugi sums with a strictly right-veering open book, if the plumbing region of the original open book veers to the left. We also provide examples of open books in S^3 demonstrating that primeness is not necessarily preserved under essential Murasugi sum, in fact not even under stabilizations a.k.a. Hopf plumbings. Furthermore, we find that trefoil plumbings need not preserve primeness. In contrast, we establish that figure-eight knot plumbings do preserve primeness.