Braid Groups

BraidGroup(W): GrpFPCox → GrpFP, Map

The braid group \(B\) 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 \(A_n\) with normalisation can be constructed with the command BraidGroup(n+1) (see Chapter ChapGrpBrd).

PureBraidGroup(W): GrpFPCox → GrpFP, Map

Returns the pure braid group of 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.

Example: Braid Groups

> 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