**MSC:**- 57M25 Knots and links in $S^3$, {For higher dimensions, See

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.