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

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Hair braid

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

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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.

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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:

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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).

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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.

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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.

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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!

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Made up by smooshing generators 1 and 2
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Made up by smooshing generator 3 with itself 3 times
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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.

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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

 

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