 
Summary: GEOMETRIC PRESENTATIONS FOR
THE PURE BRAID GROUP
DAN MARGALIT AND JON MCCAMMOND
Abstract. 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. Mo
tivated 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 [5] gave a more symmetrical presentation for the braid group on a larger
generating set that has become fashionable of late (see, for example, [4], [7], [8], or
[11]). 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 com
