# A Correspondence Between Braids and Arcs

**Figure 1.**

**Figure 2.**

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).

**Figure 3.**

**Figure 4.**

**Figure 5.**

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.

**Figure 6.**

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.

**Figure 7.**

**Figure 8.**

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.

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