Sufficient conditions for a closed n-braid (Formula Presented) to have infinite sets (Formula Presented) and (Formula Presented) are given, where (Formula Presented) denotes the set of all closed (n − 1)-braids that are obtained from (Formula Presented) via Markov destabilization, while (Formula Presented) denotes the set of all closed (n+1)-braids that are obtained from (Formula Presented) via Markov stabilization. New integer-valued conjugacy invariants for the braid group are introduced.