At a recent morning coffee meeting, I was idly playing with my hair when this was noticed by a couple of other people. This led to a discussion of different braiding styles and, because we were mathematicians, a discussion of braid theory. I continued to spend a lot of time reading about it. (Nerd-sniped.)
I didn’t know much about braid theory (or indeed group theory) before, but it turned out to be a very rich subject. I remember being introduced to group theory for the first time and finding it very hard to visualise abstract objects like generators, commutators, conjugates or normal subgroups. Braid groups may be a very useful way of introducing these: they can be demonstrated very visually and hands-on.
Suppose you have n finite strands of string. A braid is a pattern formed by crossing the strings over and under each other in a certain order, and fixing them at both ends: imagine the strings on a guitar being crossed around each other. Of course, two such patterns can be composed with each other associatively. (A plait of hair consists of such a pattern being repeatedly applied.) Each braid can be inverted by crossing the strings in the opposite order, and the identity braid is the default with no crossings (a ponytail, or a normal guitar fretboard). The set of braids therefore forms a group called Bn, the braid group on n strings. It has some similarities to the symmetric group Sn, but, as an infinite group, Bn has more structures.
Any braid can be specified by crossings of adjacent strings, which are said to generate all the braids. For i = 1, 2, …, n−1, let σi denote the act of crossing string i+1 over string i. Clearly the inverse σi−1 is to cross string i over string i+1: these two operations cancel each other out. Repeated applications of σi amounts to twirling two neighbouring strands around each other.
If you have n ≥ 4 strings, then anything you do with strings 1 and 2 will have no effect on strings 3 and 4, and vice-versa. In general, the transpositions σi and σj commute with each other provided that i and j are at least 2 apart. However, neighbouring transpositions do not compose with each other, and in fact the commutator between σ1 and σ2 is
[σ1, σ2] = σ1σ2σ1−1σ2−1
We have only got transpositions of neighbouring braids at the moment, but as mentioned above these are enough to create all other patterns. The basic tool is conjugation. On 3 strands, the conjugation
first takes strand 2 over strand 1. It then weaves strand 3 over the new strand 2 (originally strand 1). Then it takes strand 1 over strand 2. The total effect is that strand 3 goes over strand 1, while strand 2 goes over both of them. It is as though we applied σ2 after renaming strands 1 and 2 with each other: hence the mantra conjugation is relabelling (and that’s also why we see it in changes of bases in linear algebra). Note that this conjugation leaves a strand hanging loose, even though it appears to be involved in the pattern: this is a danger of the neighbouring transpositions notation.
Clearly, Bn admits subgroups where you take some consecutive subset of the strings and braid only them. (There are other subgroups as well.) Suppose n ≥5 and H is the subgroup from braiding only strands 1, 2 and 3, i.e. the subgroup generated by σ1 and σ2. These commute with σj for j ≥4; those generate the centraliser of H.
The braid group Bn has a surjective homomorphism onto the symmetric group Sn, since a braid is a permutation of the strings. The kernel of this homomorphism consists of all those braids which eventually leave the strings in their original order.
The most basic plait (other than the unplait, or ponytail) is on n = 3 strands of hair, and consists of repetitions of the element σ1σ2−1.
A more complicated pattern weaves together n = 4 strands of hair, with the basic element as σ1σ2−1σ3. The generalisation of this to higher n is an exercise for the reader. A fishtail braid is also on 4 strands, but it has a different repeating element:
However, not all braids are created equal. For example, the braid with repeating element
does not make a very good hairstyle: it is merely a twisted ponytail. A stable hairstyle should involve these transpositions as well as their inverses: i.e. a strand should be made to go over, under, over, under. A strand should be used often, to avoid it hanging loose.
The braid group is a non-abelian infinite group. In a later post, I may talk about the application of such a group to cryptography.