Braid group

Summary

The braid group $B_n$ is the group of braids with $n$ strands. Two braids are equal when one can be deformed into the other, and the product places one braid after the other. Artin gave its presentation with generators $\sigma_1, \ldots, \sigma_{n-1}$, the crossings of neighbouring strands, so $B_n$ is an Artin group. It is also the fundamental group of the space of sets of $n$ distinct points in the plane. It is also the mapping class group of a disk with $n$ marked points. Braid groups link algebra with knot theory: by Alexander's theorem, every link is the closure of a braid.