Braid group

From Groupprops
Jump to: navigation, search

Definition

In terms of the presentation using Artin braid relations

The braid group on n letters, denoted B_n, is defined as follows:

\langle s_1, s_2, \dots, s_{n-1} \mid s_is_{i+1}s_i = s_{i+1}s_is_{i+1} \ \forall \ 1 \le i \le n - 2, s_is_j = s_js_i \ \forall \ |i - j| > 1 \rangle.

Facts

There is a natural surjective homomorphism from the braid group B_n to the symmetric group S_n, that sends each s_i to the transposition (i,i+1) in S_n. One way of seeing this is noting that the presentation of S_n is obtained by tacking on more relations (namely, the relations that each s_i square to the identity) to the relations for B_n.

The kernel of this homomorphism is the pure braid group and is denoted P_n. P_n is thus a normal subgroup of finite index in B_n. The index is n!.

Particular cases

Value of n Value of n - 1 (number of generators for the Artin presentation) Braid group B_n Symmetric group S_n Pure braid group P_n(kernel of natural homomorphism to symmetric group)
1 0 trivial group trivial group trivial group
2 1 group of integers cyclic group:Z2 group of integers
3 2 braid group:B3 symmetric group:S3 pure braid group:P3
4 3 braid group:B4 symmetric group:S4 pure braid group:P4
5 4 braid group:B5 symmetric group:S5 pure braid group:P5