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.