By the Classification Theorem of closed \$3\$-braids given by J. Birman and W. Menasco, it is known that there are only finitely many mutually non-conjugate \$n\$-braids (\$n= 1, 2\$ or \$3\$) having the same closure.
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.