Positive presentations of the braid groups and the embedding problem

Abstract

We determine when a set of conjugate braids generates the n-braid group. A set of conjugate braids corresponds to a graph of n vertices that is immersed in the plane and so it generalizes the presentations given in [4,10,12]. We investigate when such a presentation becomes a positive presentation and when such a positive presentation has the property that the semigroup of positive words embeds in the n-braid group.