Braid structures in knot complements, handlebodies and 3-manifolds · arXivDesk
math/0008235Aug 31, 200016 pages, 4 figures, to appear in the proceedings of Knots in Hellas '98, Series of Knots and Everything, Vol. 24, World Scientific
Braid structures in knot complements, handlebodies and 3-manifolds
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 Bm,n. Via concatenation Bm,n
Nearby in the stack
acquires a group structure. The objective of this paper is to find a presentation for
Bm,n
using the structure of its corresponding pure braid subgroup,
Pm,n
, and the fact that it is a subgroup of the classical Artin group
Bm+n
. Then we give an irredundant presentation for
Bm,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.