Braid group: Difference between revisions
No edit summary |
No edit summary |
||
| Line 6: | Line 6: | ||
<math>\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</math>. | <math>\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</math>. | ||
==Particular cases== | |||
{| class="sortable" border="1" | |||
! Value of <math>n</math> !! Value of <math>n - 1</math> (number of generators for the Artin presentation) !! [[Braid group]] <math>B_n</math> !! [[Symmetric group]] <math>S_n</math> !! [[Pure braid group]] <math>P_n</math>(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]] | |||
|} | |||
| 3 || 2 || | |||
Revision as of 19:23, 6 October 2010
Definition
In terms of the presentation using Artin braid relations
The braid group on letters, denoted , is defined as follows:
.
Particular cases
| Value of | Value of (number of generators for the Artin presentation) | Braid group | Symmetric group | Pure braid group (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 |
| 3 || 2 ||