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!

A correspondence between braids and arcs

I’ve been thinking a lot about the braid group recently, and different ways of studying it. You can read my original post on the braid group here. The arcs I will talk about in this post have really taken over my PhD in the past few weeks, so much so that I’ve even started drawing them on beer mats in the pub!

There is a rather nice correspondence between braids in the braid group and arcs on a punctured disc (think pancake with a few little holes in the middle). In this post I will try and explain this correspondence in the case of the three strand braid group. In this instance we have to imagine a disc with 3 punctures (little holes):
punctured-disc

We will start of with the arc that corresponds to the identity braid, which we draw from the centre of the bottom of the disc to the left puncture.

identity-2

Below we show the identity arc on the left and the identity braid on the right.

identity

So what happens when we have a braid with a twist, like the one below? How does the arc have to change to incorporate information about this braid?

braid

Consider the braid: it has three strands and we have three punctures! The left strand crosses over the middle strand and we can incorporate this into the picture by starting with the identity arc and letting our left puncture and middle puncture rotate around each other to swap places, with the left one taking the upper path (so a clockwise rotation). We let the arc follow the puncture as though it is attached, making sure it does not intersect itself or the other punctures. This process is pictured in the two images below.

arc

single-braid

For every braid in the braid group (not just the simple ones with one crossing) an arc can be drawn. When you smoosh (see the braid group post) some simpler braids together to make a longer braid, you can draw the corresponding arc one step at a time, following the braid in a downwards direction to find out the next step you should take.

Below is an image of the arc changing as the braid grows longer. At the first crossing the arc moves over the left puncture as before but then there is a second crossing where now the middle strand of the braid moves under the right strand and therefore the middle puncture will rotate with the right one. This time the middle one moves under since the middle strand moves under on the braid, giving us an anticlockwise rotation.

time-series

Below is an example of a more complicated arc. If you want a fun challenge you could try to draw out the braid that it is related to!

twisty-arc

You might wonder what use it is to draw these arcs when we already know how to work with the braids. One reason is that we can learn more about certain braids by drawing the corresponding arcs and recognising patterns or algorithms that weren’t obvious from studying the braids. This technique is used quite a lot in mathematical proof, where an answer may be a lot simpler to see when the question is formulated in a different way (for example with arcs instead of braids).

I’ll end with a couple of pictures from my ‘everyday life’. Here is a wee set up I put together to help me figure out the twists and turns for some arc examples I was working on (a hard mornings work…)

string

And finally here is a shot of my current blackboard, a mess of arc drawing in progress!

blackboard