Moreover they prove that if there is infinitely many mutually non-conjugate $n$-braids having the same closure, then all but finitely many of them are related by exchange moves.

H. Morton discovered an infinite sequence of pairwise non-conjugate $4$-braids whose closures are equivalent to the unknot and E. Fukunaga gave an infinite sequence of pairwise non-conjugate $4$-braids whose closures are equivalent to the $(2,k)$-torus link for any $k$.

For any $n$-braid $b$ $(n \ge 3)$ whose closure is a knot, we give an infinite sequence of pairwise non-conjugate $(n+1)$-braids which have the same closures as $b$ and we show that the closures of the braids in our sequence fall into a single equivalence class by exchange moves.