Split braids

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 $B_{n - 1}^{n - 1}$.