Satellite fully positive braid links are braided satellite of fully positive braid links

TL;DR

This paper characterizes fully positive braid links that are satellite links, showing they are satellites of other fully positive braid links with patterns containing many full twists, and uses this to characterize the unknot.

Contribution

It provides a necessary and sufficient condition for fully positive braid links to be satellite links, linking their structure to patterns with many full twists.

Findings

Fully positive braid links are satellite links iff they are satellites of certain fully positive braid links.

The pattern in the satellite construction must contain sufficiently many full twists.

The unknot can be characterized by properties of certain braided satellites.

Abstract

A link in $S^{3}$ is a fully positive braid link if it is the closure of a positive braid that contains at least one full-twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully positive braid link $C$ such that the pattern is a positive braid that contains sufficiently many full twists, where the number of necessary full twists only depends on $C$. As an application, we give a characterization of the unknot by the property that certain braided satellite is a (fully) positive braid knot.