Braid structures in knot complements, handlebodies and 3-manifolds

Details Braid structures in knot complements, handlebodies and 3-manifolds Braid structures in knot complements, handlebodies and 3-manifolds

2000-08-31

arxiv.org/abs/math/0008235v1

{\it Proceedings of Knots in Hellas '98}, World Scientific Press, Series of Knots and Everything {\bf 24}, 274--289 (2000).

Mathematics

Geometric Topology

16 pages, 4 figures, to appear in the proceedings of Knots in Hellas '98, Series of Knots and Everything, Vol. 24, World Scien

Scientific paper

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.

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