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Proceedings of the American Mathematical Society

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

 

 

Split braids


Author: Stephen P. Humphries
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
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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}$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1072340-1
Keywords: Braid group, algorithm
Article copyright: © Copyright 1991 American Mathematical Society