# A non-degenerate exchange move always produces infinitely many non-conjugate braids

## TL;DR

This paper proves that if a link has a closed braid with a non-degenerate exchange move, it must have infinitely many non-conjugate braid representations, revealing complex structure in braid conjugacy classes.

## Contribution

It establishes that non-degenerate exchange moves on closed braids imply infinitely many non-conjugate braid representations of the link.

## Key findings

- Non-degenerate exchange moves lead to infinitely many non-conjugate braids.
- Links with such moves have complex conjugacy class structures.
- The result applies to closed n-braids representing links.

## Abstract

We show that if a link $L$ has a closed $n$-braid representative admitting non-degenerate exchange move, an exchange move that does not obviously preserve the conjugacy class, $L$ has infinitely many non-conjugate closed $n$-braid representatives.