The braid groupB of the Coxeter group W as a finitely presented group,
together with the natural map W to B.
Words in the
braid group are not automatically normalised.
However, the braid group of type An with normalisation can be constructed
with the command BraidGroup(n+1) (see Chapter BRAID GROUPS).
Returns the pure braid groupof the Coxeter group W, ie. the kernel of
the epimorphism from the braid group of W to W.
Words in the pure braid group are not automatically normalised.
> W<a,b,c> := CoxeterGroup(GrpFPCox, "B3");
> W;
Coxeter group: Finitely presented group on 3 generators
Relations
a * b * a = b * a * b
a * c = c * a
(b * c)^2 = (c * b)^2
a^2 = Id($)
b^2 = Id($)
c^2 = Id($)
> B<x,y,z> := BraidGroup(W);
> B;
Finitely presented group B on 3 generators
Relations
x * y * x = y * x * y
x * z = z * x
(y * z)^2 = (z * y)^2
> P := PureBraidGroup(W);
> P;
Finitely presented group P on 3 generators
Generators as words in group B
P.1 = x^2
P.2 = y^2
P.3 = z^2
V2.28, 13 July 2023