On normalizers of colored braids
\(N(A)\) denotes the normalizer of a braid \(A\) in the group \(B_{n+1}\) of \((n+1)\)-string braids, and \(N^ k(A)\) its normalizer in the group \(K_{n+1}\) of ``colored'' braids with trivial permutation. A colored braid is \(j\)-pure, if it becomes trivial after removing its \(j\)-th string. Normalizers of pure braids in \(K_{n+1}\) were determined by the reviewer [in Abh. Math. Semin. Univ. Hamb. 27, No. 1-2, 97-115 (1964; Zbl 0134.431)]. Under certain conditions \(N^ k(A)\) has a ``standard'' form: \(\{\Delta^ 2\} \cdot \{A\}\); here \(\{\Delta^ 2\}\) is the (cyclic) center of \(B_{n+1}\) resp. \(K_{n+1}\), and \(\{A\}\) the cyclic group generated by \(A\). The author proves Theorem 1: If \(A\) is a colored braid, then \(N(A) = N(A^ m)\), \(m \neq 0\). Theorem 3 gives a simple condition in terms of the linking numbers of pairs of strings for \(N(A)\) to take the standard form \(\{\Delta^ 2\}\cdot \{A\}\). In Theorem 2, on the other hand, for a certain class of pure braids properties are deduced in case \(N(A)\) is not standard. The proofs use the fact that \(B_{n+1}\) is torsion free; this was proved by Fadell and Neuwirth in 1962, not by E. Artin in 1926.
scientific article; zbMATH DE number 525664
scientific article