Braid pictures for Artin groups

We dene 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 an , bn = cn and dn and the aÆne diagrams An , Bn , Cn and Dn as subgroups of the braid groups of various simple orbifolds. The cases dn , Bn , Cn and Dn are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except An the quotient is nite. We illustrate the value of our braid calculus by performing with pictures a nontrivial calculation in the Artin groups of type dn . 1 Introduction The purpose of this paper is to establish and explain a very close connection between two quite dierent-seeming generalizations of the classical braid group. One generalization is to the class of Artin groups. This generalization is very natural when one studies the braid group from the point of view of singularity theory, or Lie theory, or the theory of reection group..