Figure 2:
Worldlines for N indistinguishable particles. (a) Elementary braids involve interchanging two adjacent strands, passing the higher over (out of page) the lower. An important theorem states that any braid can be obtained by multiplying elementary braids and their inverses. (b) There is a nonobvious relation among products of braids, known as the Yang-Baxter relation, our Eq. (2). Its geometric content is displayed here. The upper and lower braids, respectively, are the left- and right-hand sides of Eq. (2). It is easy to see that these two braids can be continuously related, i.e., that they are topologically equivalent.