Today we are going to learn about braid groups! But before we jump in, let’s do a quick recap of what we have covered so far. Recall that a group is a set \(G\), together with a binary operation \(\bullet\), such that the following group axioms hold:
(Associativity) For every \(a,b,c\in G\), \((a\bullet b)\bullet c = a\bullet (b\bullet c)\).
(Identity element) There exists \(\text{id}_G\in G\) such that for all \(a\in G\), \(\text{id}_G\bullet a = a = a \bullet\text{id}_G\).
(Inverse element) For every \(a\in G\), there exists \(a^{-1}\in G\) such that \(a\bullet a^{-1} = \text{id}_G = a^{-1}\bullet a\).
So far we have seen two main examples of groups. The first being symmetry groups, and the second being symmetric groups. Today we are going to look at another class of groups called braid groups!
Like most groups, the study of braid groups has applications all over mathematics in areas such as topology, knot theory, physics, fluid mechanics, and algebraic geometry. Besides it’s far reaching applications, braid groups are also visually nice and fun to play with because the elements in the underlying set of a braid group actually look like braids! Recall again that a group is a set together with a binary operation. Together, we will learn about the underlying set of a braid group. And throughout various exercises, you will figure out what the binary operation is!
The underlying set of a braid group is a certain collection of objects called braids. As hinted at earlier, these objects do in fact look like the braids you see in your everyday life. Here are some examples of braids out in the wild: