Editing Braid group (section)

==Introduction==
In this introduction let {{math|''n'' {{=}} 4}}; the generalization to other values of {{math|''n''}} will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a ''braid''. Often some strands will have to pass over or under others, and this is crucial: the following two connections are ''different'' braids:
:
{| valign="centre"
|-----
| [[File:braid s1 inv.png|The braid sigma 1<sup>−1</sup>]]
| &nbsp;&nbsp;&nbsp;is different from&nbsp;&nbsp;&nbsp;
<td>[[File:braid s1.png|The braid sigma 1]]
|}

On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered ''the same'' braid:
:
{| valign="centre"
|-----
| [[File:braid s1 inv.png|The braid sigma 1<sup>−1</sup>]]
| &nbsp;&nbsp;&nbsp; is the same as&nbsp;&nbsp;&nbsp;
<td>[[File:braid s1 inv alt.png|Another representation of sigma 1<sup>−1</sup>]]
|}

All strands are required to move from left to right; knots like the following are ''not'' considered braids:
:
{| valign="centre"
|-----
| [[File:braid nobraid.png|Not a braid]]
<td>&nbsp;&nbsp;&nbsp;is not a braid
|}

Any two braids can be ''composed'' by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:
:
{| valign="centre"
|-----
| [[File:braid s3.png]]
| &nbsp;&nbsp;&nbsp; composed with &nbsp;&nbsp;&nbsp;
| [[File:braid s2.png]]
| &nbsp;&nbsp;&nbsp; yields &nbsp;&nbsp;&nbsp;
<td>[[File:braid s3s2.png]]
|}

Another example:
:
{| valign="centre"
|-----
| [[File:braid s1 inv s3 inv.png]]
| &nbsp;&nbsp;&nbsp; composed with &nbsp;&nbsp;&nbsp;
| [[File:braid s1 s3 inv.png]]
| &nbsp;&nbsp;&nbsp; yields &nbsp;&nbsp;&nbsp;
| [[File:braid s3 inv squared.png]]
|}
The composition of the braids {{math|σ}} and {{math|τ}} is written as {{math|στ}}.

The set of all braids on four strands is denoted by <math>B_4</math>. The above composition of braids is indeed a [[group (mathematics)|group]] operation. The [[identity element]] is the braid consisting of four parallel horizontal strands, and the [[inverse element|inverse]] of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre. (The first two example braids above are inverses of each other.)