The $n$th root of a braid is unique up to conjugacy

Abstract

We prove a conjecture due to Makanin: if $\alpha$ and $\beta$ are elements of the Artin braid group $B_n$ such that ${\alpha }^k={\beta }^k$ for some nonzero integer $k$, then $\alpha$ and $\beta$ are conjugate. The proof involves the Nielsen–Thurston classification of braids.