Braid pictures for Artin groups

We de ne the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A(n), B-n = C-n and D-n and the affine diagrams (A) over tilde (n), (B) over tilde (n), (C) over tilde (n) and (D) over tilde (n) as subgroups of the braid groups of various simple orbifolds. The cases D-n, (B) over tilde (n), (C) over tilde (n) and (D) over tilde (n) are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except (A) over tilde (n) the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type D-n.