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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Split braids
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by Stephen P. Humphries
Proc. Amer. Math. Soc. 113 (1991), 21-26
DOI: https://doi.org/10.1090/S0002-9939-1991-1072340-1

Abstract:

Let ${B_n}$ be the group of braids on $n$ strings with standard generators ${\sigma _1}, \ldots ,{\sigma _{n - 1}}$. For $i \in \{ 1,2, \ldots ,n - 1\}$ we let $B_n^i$ be the subgroup of ${B_n}$ generated by the elements ${\sigma _1}, \ldots ,{\sigma _{i - 1}},{\sigma _{i + 1}}, \ldots ,{\sigma _{n - 1}}$. In this paper we give an algorithm for deciding if, given $\alpha \in {B_n}$ there is $i \in \{ 1,2, \ldots ,n - 1\}$ such that $\alpha$ is conjugate into $B_n^i$. We call such a braid a split braid. Such a split braid gives rise to a split link. This algorithm gives a partial solution to the problem of finding braids that represent reducible mapping classes. It also represents a contribution to the algebraic link problem and it gives a way of determining if a braid in ${B_n}$ can be conjugated into the subgroup ${B_{n - 1}}$, which we identify with $B_{n - 1}^{n - 1}$.
References
  • Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
  • F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235–254. MR 248801, DOI 10.1093/qmath/20.1.235
  • Wilhelm Magnus, Braid groups: a survey, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 463–487. MR 0353290
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Bibliographic Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 113 (1991), 21-26
  • MSC: Primary 20F36; Secondary 57M25
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1072340-1
  • MathSciNet review: 1072340