braidgroep

The braid group B_n, in Dutch mathematics called the braidgroep, consists of braids on n strands in three-dimensional space, with the group operation given by concatenation. An element is an equivalence class of braids with fixed endpoints, modulo ambient isotopy that preserves the endpoints.

A standard presentation uses generators sigma_1, ..., sigma_{n-1}, with relations sigma_i sigma_j = sigma_j sigma_i for |i - j| > 1 and sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} for i = 1, ..., n-2. The natural map B_n -> S_n sends a braid to the permutation of its strand endpoints; its kernel is the pure braid group P_n.

Braid groups have several important realizations and connections. They are the mapping class group of the n-punctured disk and equivalently the fundamental group of the configuration space of n ordered points in the plane. They form an Artin group of type A_{n-1} and are often studied via their combinatorial and geometric structure.

Applications and theory: braid groups underpin much of knot theory, since every link is the closure of a braid (Alexander’s theorem), and two braids define the same link precisely when related by Markov moves (Markov’s theorem). They admit various representations into linear groups, including the Burau representation and the Lawrence–Krammer–Bigelow representation, the latter establishing the linearity of B_n for all n.

For n ≥ 3, B_n is infinite and non-abelian; its center is trivial, while B_2 is isomorphic to the infinite cyclic group. The braidgroep thus occupies a central place in algebra, topology, and mathematical physics.