Geodesic
On the number of closed braids obtained as a~result of single stabilizations and destabilizations of a~closed braid
Algebra i analiz, Tome 18 (2006) no. 6, pp. 205-218.

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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. 
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A. V. Malyutin. On the number of closed braids obtained as a~result of single stabilizations and destabilizations of a~closed braid. Algebra i analiz, Tome 18 (2006) no. 6, pp. 205-218. http://geodesic.mathdoc.fr/item/AA_2006_18_6_a3/

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