Abstract
Crossings are suggested as they are in a picture of a highway overpass on a map. The identity braid has a canonical representation in which two strands never cross. Multiplication of braids is by juxtaposition, concatenation, isotopy, and rescaling. This chapter discusses Artin's braid group, Bn and its role in knot theory. The chapter illustrates ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. Artin's braid group is naturally isomorphic to the mapping class group of an n-times punctured disc. The chapter also illustrates the topological concept of a braid and of a group of braids via the notion of a configuration space. It then outlines the new developments in the area of Bn mapping, thus illustrating how to pass from diffeomorphisms to geometric braids and back again..
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