Exchange moves and braid representations of links

TL;DR

This paper demonstrates that iterated exchange moves can generate infinitely many non-conjugate braids, implying that many knots have infinitely many conjugacy classes of braid representations under certain conditions.

Contribution

It establishes a general condition under which exchange moves produce infinitely many non-conjugate braid representations for knots.

Findings

Iterated exchange moves lead to infinitely many non-conjugate braids.

Every knot has infinitely many conjugacy classes of braid representations if it admits an exchange move.

The result connects exchange moves with the diversity of braid representations of knots.

Abstract

We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.