# Definition:Braid Group

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## Description

The **braid group** is a group that has a intuitive geometric interpretation as a number of strands, where the group operation on these strands is to intertwine them.

## Generators

The generators of the braid group are elements $\sigma_i$, which intertwine strands $i$ and $i+1$ in such a way that strand $i$ runs above strand $i+1$.

## Definition

The braid group on $n$ strands is generated by $\sigma_1, \sigma_2, \ldots, \sigma_{n-1}$ and the following relations:

- $\sigma_i \sigma_j = \sigma_j\sigma_i, \forall i, j: |i-j| \ge 2$
- $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \forall i \in \{ 1, 2, \ldots, n-2 \}$

## Examples

Generator $\sigma_i$ and the inverse generator $\sigma_i^{-1}$ acting on strands $s_i, s_{i+1}$:

Relation 1. can be pictured like this:

Relation 2. can be pictured like this:

This needs considerable tedious hard slog to complete it.In particular: Some explanatory notes need filling in here: proof that it's a group, proof that the generators are actually generators, etc. all of which need to be linked to.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |