Geodesic
Systems of generators for centralizers of rigid elements of the braid group
Izvestiya. Mathematics , Tome 31 (1988) no. 2, pp. 223-244.

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The problem of describing centralizers of elements of the braid group was posed by Artin in 1947. An element of the braid group $\mathfrak B_{n+1}$ is said to be rigid if it can be represented as a positive word that is not equal to any other word in the braid semigroup. Explicit expressions are given for finite systems of generators for the centralizers of a wide class of rigid elements. The article is a continuation of the author's paper Systems of generators for the normalizers of certain elements of the braid group (Izv. Akad. Nauk SSSR. Ser. Mat., 1984, V. 48, № 3, P. 476–519), where the history of the problem is covered, and a list of references provided. Bibliography: 2 titles. 
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G. G. Gurzo. Systems of generators for centralizers of rigid elements of the braid group. Izvestiya. Mathematics , Tome 31 (1988) no. 2, pp. 223-244. http://geodesic.mathdoc.fr/item/IM2_1988_31_2_a0/

[1] Gurzo G. G., “Sistemy obrazuyuschikh dlya normalizatorov nekotorykh elementov gruppy kos”, Izv. AN SSSR. Ser. matem., 48:3 (1984), 476–519 | MR | Zbl

[2] Artin E., “Theory of Braids”, Ann. Math., 48 (1947), 101–126 | DOI | MR | Zbl