To make the braid, one dangles the strands vertically and weaves them downward however they like. Columbia University mathematician Joan Birman helped to popularize the question about 60 years ago. It has long been known that, for the first three strands of a braid, the matrices are always faithful, meaning there is a one-to-one relationship among them. In the 1970s Birman wrote a foundational book for mathematicians, Braids, Links, and Mapping Class Groups. The disks, which are defined as the insides of the loops, tell you how to calculate the Burau matrices.