Combing braids

I’m going to a conference next week, and it’s all about braids! So I thought I would write a wee post on combing, a technique which dates back to Artin in the 1940s. In fact the paper where he introduces the concept of combing finishes with the following amusing warning:

“Although it has been proved that every braid can be deformed into a similar normal form the writer is convinced that any attempt to carry this out on a living person would only lead to violent protests and discrimination against mathematics. He would therefore discourage such an experiment.” – Artin 1946

but I really don’t see it as so bad!

Combing is a technique for starting with any braid (see my introductory post on braids here) and ending up with a braid in which first the leftmost strand moves and the others stay put, then the next strand moves while the rest stay put etc etc. It’s much nicer to show this in pictures.

We want to start with any old braid, say this one:

original

 

and transform it into a braid where the strands move one at a time, like the following one. I’ve coloured the strands here so you can see that, reading the braid from top to bottom, first the red strand moves (i.e. all crossing involve the red strand, until it is finished), and then the green, and then the blue.

final coloured

 

For convenience I’ll only look at braids called pure braids, where each strand starts and ends at the same position. You can easily comb non-pure braids, you just need to add an appropriate twist right at the end to make them finish in the correct positions.

So how do we do this? Consider the first strand, I’ve coloured it red to make it clear. We want all the crossings between red and black strands to happen before (higher up than) a crossing of two black strands. So in this case the crossing circled in yellow are okay, because they happen lower down than any crossing involving the red strand. The crossings circled in blue and green need to be changed.

strand 1 highlight

 

We can slide some crossings of black strands down past the red and black crossings, as they don’t interfere. Here we can do it with the crossing circled in blue, as shown:

strand 1 step 1

 

We can start to do it with the crossing circled in green, but we encounter a problem as it wont simply slide past the red strand crossing below it. Moving this crossing down requires using some of the braid relations (see braid post) to replace a few crossings with an equivalent section in which the red strand moves first, as follows:

strand 1 step 2

Even though this braid looks different than the previous one they are in fact the same (you can always test this with string!). Now we have a braid in which the first strand moves before any others. Since all the first stand action is now at the top of the braid, we can now ignore the first strand all together, and consider the rest of the braid, as show below:

strand 1 forgetting

 

we only need to consider the following section now, and again we can put this into a form where only the first strand moves.

strand 2 beginning

In this case using braid relations gives us the following:

strand 2 final

And we can now ignore the green strand!

strand 2 forgetting

Colouring the first strand in this final section gives us no crossing that don’t involve the first strand:

strand 3 start

and we colour the last strand yellow for fun!

strand 3

Remembering all the pieces we have ignored gives us the full combed braid, where we focus on the leftmost strand until it ‘runs out of moves’ before looking to the next one.

final coloured

And this is exactly the same as the original braid, which looks a lot messier when coloured:

original coloured

Why might we want to do this? In some cases it makes mathematical proofs a lot easier. For me, recently I have been focusing only on what the first strand is doing, and so I want a technique to push the other strands down and away!

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The braid group

Apologies for the delay in writing this post. Sometimes when one does maths everyday the last thing they feel like doing when they get home is writing about maths, and I hope that’s enough of an excuse.

This post is going to focus on braid groups, what they are and how we can visualise them. The braid group is the most popular/simplest example of an Artin group, and I guess in some sense my whole PhD is on Artin groups.

So what is the braid group? We have all heard of hair braids

IMG_8730
Hair braid

And the hair braid is an example of a braid in the braid group on 3 strands

IMG_8729
An element of the braid group on three strands, with the same crossover pattern as a hair braid.

A group can be thought of as a collection of elements with some sort of operation which for the purpose of the post we will call smooshing i.e. you can smoosh two elements together to get a new element in the group. There is an identity element which doesn’t do anything when you smoosh it with other elements and each element has an inverse element: when you smoosh an element and its inverse you get the identity element back.

SO what is the braid group? The braid group is defined on a set number of strands. An element in this group looks like you have taken these strands and neatly lain them out, then twisted them together in some way. Here are some elements in the braid group on 4 strands, where we have drawn the picture such that if one strand passes over the top of another, the bottom strand seems to break.

IMG_8728

 

IMG_8726

 

IMG_8727

SO what is the smooshing relationship between these elements? If we want to smoosh together two braids we simply tie together the bottom of one braid with the top of the other, like so:

IMG_8725
Smooshing of two elements in the 4 strand braid group

The identity element is the braid where no strands are twisted, as tying this to any other braid doesn’t change it (it makes it a bit longer but we don’t care about stretching and squashing because we are doing topology).

IMG_8724
Smooshing of an element in the 4 strand braid group with the identity

And here is an example of a braid and its inverse: when you smoosh them together you get the identity element.

IMG_8723
Smooshing of a 4-strand braid and its inverse element

Okay so we have a few cool pictures and we are starting to understand what the braid group is, but if we want to do some maths then we had better be able to write down the elements. And writing down a complicated twisty things might get tricky! A solution to this problem is to work with generators of the group. We can get any element of the braid group on 4 strands by simply working with these 6 elements and smooshing them together on various ways. Notice that the 6 elements are in fact three elements and their inverses.

IMG_8722
6 generators of the braid group on 4 strands

If we give them all names (or in this case numbers) then we can consequently write down any element by saying the order we have to smoosh these 6 generators together to get it. So we are ready to do some maths!

IMG_8728
Made up by smooshing generators 1 and 2
IMG_8726
Made up by smooshing generator 3 with itself 3 times
IMG_8727
Made up by smooshing generators 1, 4 and 5 with 1 on the top and 5 on the bottom

What I’m actually trying to do involves the classifying space of the braid group and fitting braid groups inside each other by adding an extra straight strand to elements.

IMG_8732
An element of the 4 strand braid group being written as a 5 strand braid in the 5 strand braid group, by adding an extra strand