SPLIT BRAIDS

Let B(n) be the group of braids on n strings with standard generators sigma-1, ... , sigma-n-1. For i is-an-element-of {1, 2, ..., n-1} we let B(n)i be the subgroup of B(n) generated by the elements sigma-1, ..., sigma-i-1, sigma-i+1, ..., sigma-n-1. In this paper we give an algorithm for deciding if, given alpha is-an-element-of B(n) there is i is-an-element-of {1, 2, ..., 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.