group theory

# Contents

## Idea

The braid group ${\mathrm{Br}}_{n}$ is the group whose elements are isotopy classes of $n$ 1-dimensional braids running vertically in 3-dimensional Cartesian space, the group operation being their concatenation.

Here a braid with $n$ strands is thought of as $n$ pieces of string joining $n$ points at the top of the diagram with $n$-points at the bottom.

(This is a picture of 3 strand braid.)

We can transform / ‘isotope’ these braid diagrams just as we can transform knot diagrams, again using Reidemeister moves. The ‘isotopy’ classes of braid diagrams form a group in which the composition is obtained by putting one diagram above another.

The identity consists of $n$ vertical strings, so the inverse is obtained by turning a diagram upside down:

This is the inverse of the first 3-braid we saw.

There are useful group presentations of the braid groups. We will return later to the interpretation of the generators and relations in terms of diagrams.

## A presentation of ${\mathrm{Br}}_{n+1}$

The Artin braid group, ${\mathrm{Br}}_{n+1}$, defined using $n+1$ strands is a group given by

• generators: ${y}_{i}$, $i=1,\dots ,n$;

• relations:

• ${r}_{i,j}\equiv {y}_{i}{y}_{j}{y}_{i}^{-1}{y}_{j}^{-1}$ for $i+1

• ${r}_{i,i+1}\equiv {y}_{i}{y}_{i+1}{y}_{i}{y}_{i+1}^{-1}{y}_{i}^{-1}{y}_{i+1}^{-1}$ for $1\le i.

## Examples

We will look at such groups for small values of $n$.

###### The group ${\mathrm{Br}}_{1}$

By default, ${\mathrm{Br}}_{1}$ has no generators and no relations, so is trivial.

###### The group ${\mathrm{Br}}_{2}$

By default, ${\mathrm{Br}}_{2}$ has one generator and no relations, so is infinite cyclic.

###### The group ${\mathrm{Br}}_{3}$

(We will simplify notation writing $u={y}_{1}$, $v={y}_{2}$.)

This then has presentation

$𝒫=\left(u,v:r\equiv uvu{v}^{-1}{u}^{-1}{v}^{-1}\right).$\mathcal{P} = ( u,v : r \equiv u v u v^{-1} u^{-1} v^{-1}).

It is also the ‘trefoil group’, i.e., the fundamental group of the complement of a trefoil knot.

###### The group ${\mathrm{Br}}_{4}$

Simplifying notation as before, we have generators $u,v,w$ and relations

• ${r}_{u}\equiv vwv{w}^{-1}{v}^{-1}{w}^{-1}$,
• ${r}_{v}\equiv uw{u}^{-1}{w}^{-1}$,
• ${r}_{w}\equiv uvu{v}^{-1}{u}^{-1}{v}^{-1}$.

## References

Classical references are

• Joan S. Birman, Braids, links, and mapping class groups, Princeton Univ Press, 1974.
• R. H. Fox, L. Neuwirth, The braid groups, Math. Scand. 10 (1962) 119-126, pdf, MR150755