Geometric Presentations for the Pure Braid Group

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged “convexly”, which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the “rotation presentation”, we introduce presentations for the pure braid group that we call the “twist presentation” and the “swing presentation”. From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation. The braid group has had a standard presentation on a minimal generating set ever since it was first defined by Emil Artin in the 1920s [3]. In 1998, Birman, Ko, and Lee [7] gave a more symmetrical presentation for the braid group on a larger generating set that has become fashionable of late (see, for example, [4], [9], [10], or [13]). Our goal is to apply a similar idea to the pure braid group. The standard finite presentation for the pure braid group (also due to Artin [2]) is slightly complicated and not that easy to remember. The presentations introduced here are, we believe, simple, easy to remember and intuitively clear. The article is structured as follows. In §1 we present a variation of the Birman–Ko–Lee presentation for the full braid group that we call the rotation presentation, and in sections 2, 3, and 4 we establish increasingly simple presentations for the pure braid group that we call the modified Artin presentation, the twist presentation, and the swing presentation. For these presentations, we think of the braid group as the fundamental group of the configuration space of n points in the disk. If we reinterpret the swing presentation in terms of mapping class groups, we get a presentation where the generators are Dehn twists, and the relations are the disjointness relation and the lantern relation (see section 4). The final section explores some possible extensions. 1. Braids This section gives an unusual presentation of the full braid group using the notion of a convexly punctured disc. In addition to proving that it is equivalent to the (closely related) Birman–Ko–Lee presentation, we introduce several notions that pave the way for our new presentations of the pure braid group. The n-string braid group Braidn can be viewed as the fundamental group of the configuration space of n distinct but indistinguishable points in a disc: Braidn ∼= π1(C(D, n)). The points are called punctures and the elements of Braidn can be thought of as homotopy classes of based loops in C(D, n), or equivalence classes of Date: June 16, 2006. 1