Sofia Lambropoulou
Braid groups related to knot complements, handlebodies and 3--manifolds
Preprint series: Mathematica Gottingensis
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.

Keywords: Braid groups, pure braid groups, cosets, knot complements, handlebodies, 3-manifolds.