By Rachael Boyd
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 Ph.D. 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) (see Figure 1):
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 (see Figure 2).
Below we show the identity arc on the left and the identity braid on the right (see Figure 3).
So what happens when we have a braid with a twist, like the one to the right (see Figure 4)? How does the arc have to change to incorporate information about this 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 Figures 5 and 6.
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 (see Figure 7). 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.
To the right is an example of a more complicated arc (see Figure 8). If you want a fun challenge, you could try to draw out the braid that it is related to!
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 morning's work…)
And finally, here is a shot of my current blackboard, a mess of arc drawing in progress!
This work is republished with permission from the Picture This Maths blog.
Rachael Boyd is a Ph.D. student at the University of Aberdeen in Scotland, working in algebraic topology. Her interests lie in homological stability and group homology of Coxeter and Artin groups. She writes a blog with Anna Seigal at https://picturethismaths.wordpress.com/.