On the number of closed braids obtained as a result of single stabilizations and destabilizations of a closed braid

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