Satellite knots that cannot be represented by positive braids with full twists

TL;DR

This paper identifies satellite knots that are closures of positive minimal braids but cannot be represented with positive braids containing full twists, revealing new distinctions among Lorenz knots and addressing a longstanding question.

Contribution

It demonstrates the existence of satellite knots with positive minimal braids that defy representation with full twists, expanding understanding of braid and knot classifications.

Findings

Certain satellite knots cannot be represented by positive braids with full twists.

Infinitely many satellite knots with Lorenz pattern and companion are not Lorenz knots.

Answers a question about Lorenz knots and satellite constructions posed in the 1980s.

Abstract

A positive braid with at least one full twist is known to be a minimal braid, i.e, it achieves the braid index for its closure. In this paper we find knots that are the closure of positive minimal braids that cannot be represented by positive braids with full twists. More precisely, we show that some satellite knots with companions and patterns given as the closure of positive braids cannot be represented as the closure of positive braids with full twists. As a consequence, we find infinitely many satellite knots with companions and patterns being Lorenz knots that are not Lorenz knots. This gives an answer to the question whether a satellite knot having Lorenz pattern and companion is also a Lorenz knot, originally addressed by Birman and Williams in a special case in the 1980s.