group theory

# Contents

## Idea

The braid group $Br_n$ 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 a $3$-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.

## Geometric definition

Let $C_n \hookrightarrow \mathbb{C}^n$ be the space of configurations of $n$ points in the complex plane, whose elements are those $n$-tuples $(z_1, \ldots, z_n)$ such that $z_i \neq z_j$ whenever $i \neq j$. The symmetric group $S_n$ acts on $C_n$ by permuting coordinates. Let $C_n/S_n$ be the orbit space (the space of $n$-element subsets of $\mathbb{C}$ if one likes), and let $[z_1, \ldots, z_n]$ be the image of $(z_1, \ldots, z_n)$ under the quotient $\pi: C_n \to C_n/S_n$. We take $p = (1, 2, \ldots, n)$ as basepoint for $C_n$, and $[p] = [1, 2, \ldots n]$ as basepoint for $C_n/S_n$.

###### Definition

The braid group $Br_n$ is the fundamental group $\pi_1(C_n/S_n, [p])$. The pure braid group $P_n$ is $\pi_1(C_n, p)$.

Evidently a braid $\beta$ is represented by a path $\alpha: I \to C_n/S_n$ with $\alpha(0) = [p] = \alpha(1)$. Such a path may be uniquely lifted through the covering projection $\pi: C_n \to C_n/S_n$ to a path $\tilde{\alpha}$ such that $\tilde{\alpha}(0) = p$. The end of the path $\tilde{\alpha}(1)$ has the same underlying subset as $p$ but with coordinates permuted: $\tilde{\alpha}(1) = (\sigma(1), \sigma(2), \ldots, \sigma(n))$. Thus the braid $\beta$ is exhibited by $n$ non-intersecting strands, each one connecting an $i$ to $\sigma(i)$, and we have a map $\beta \mapsto \sigma$ appearing as the quotient map of an exact sequence

$1 \to P_n \to Br_n \to S_n \to 1$

which is part of a long exact homotopy sequence corresponding to the fibration $\pi: C_n \to C_n/S_n$.

## Group-theoretic descriptions

### Artin presentation of $Br_{n+1}$

The Artin braid group, $Br_{n+1}$, defined using $n+1$ strands is a group given by

• generators: $y_i$, $i = 1, \ldots, n$;

• relations:

• $r_{i,j} \equiv y_i y_j y_i^{-1} y_j^{-1}$ for $i+1 \lt j$

• $r_{i,i+1}\equiv y_i y_{i+1} y_i y_{i+1}^{-1} y_i^{-1} y_{i+1}^{-1}$ for $1 \leq i \lt n$.

### In terms of automorphisms on free groups

The braid group $B_n$ may be alternatively described as the mapping class group of a 2-disk $D^2$ with $n$ punctures (call it $X_n$). Meanwhile, the fundamental group $\pi_1(X_n)$ (with basepoint on the boundary) is a free group $F_n$ on $n$ generators; the functoriality of $\pi_1$ implies we have an induced homomorphism

$Aut(X_n) \to Aut(\pi_1(X_n)) = Aut(F_n).$

If an automorphism $\phi: X_n \to X_n$ is isotopic to the identity, then of course $\pi_1(\phi)$ is trivial, and so the homomorphism factors through the quotient $MCG(X_n) = Aut(X_n)/Aut_0(X)$, so we get a homomorphism

$B_n = MCG(X_n) \to Aut(F_n)$

and this turns out to be an injection.

Explicitly, the generator $y_i$ used in the Artin presentation above is mapped to the automorphism $\sigma_i$ on the free group on $n$ generators $x_1, \ldots, x_n$ defined by

$\sigma_i(x_i) = x_{i+1}, \sigma_i(x_{i+1}) = x_{i+1}^{-1} x_i x_{i+1}, \; \else\; \sigma(x_j) = x_j.$

## Examples

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

###### The group $Br_1$

By default, $Br_1$ has no generators and no relations, so is trivial.

###### The group $Br_2$

By default, $Br_2$ has one generator and no relations, so is infinite cyclic.

###### The group $Br_3$

(We will simplify notation writing $u = y_1$, $v = y_2$.)

This then has presentation

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

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

###### The group $Br_4$

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

• $r_u \equiv v w v w^{-1} v^{-1} w^{-1}$,
• $r_v \equiv u w u^{-1} w^{-1}$,
• $r_w \equiv u v u v^{-1} u^{-1} v^{-1}$.

## 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