The braid groupB of the Coxeter group W as an 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