  We consider braids on $m+n$ strands, such that the first $m$ strands are
trivially fixed.
  We denote the set of all such braids by $B_{m,n}$. Via concatenation
$B_{m,n}$ acquires a group structure. The objective of this paper is to find a
presentation for $B_{m,n}$ using the structure of its corresponding pure braid
subgroup, $P_{m,n}$, and the fact that it is a subgroup of the classical Artin
group $B_{m+n}$. Then we give an irredundant presentation for $B_{m,n}$. The
paper concludes by showing that these braid groups or appropriate cosets of
them are related to knots in handlebodies, in knot complements and in c.c.o.
3--manifolds.
