Braiding and Knots

Table of Contents

  1. The Braid Group
  2. Knots
  3. Anyons

Braids

Wikipedia article on the Braid Group

A braid can be formed by exchanging the endpoints of strings. Take for example a braid with i = 5 points. We can represent an exchange with (i-1) number of operations, σ1, σ2, σ3, σ4 . σi exchanges the i point with the i+1 point. The inverse of an exchange σi-1 is just exchanging it the opposite way.

B5 Braiding
Fig.1 - Braiding for n=5

Braiding has the same properties as a group!

  1. The Indentity is the braid with straight lines (no crossings)
  2. Any combination of braidings is also a braiding
  3. For any braiding, the inverse can be found by flipping the picture

Closure
Fig.2 - The composition of the first two Braids yields the last one

One might want to compare the Braid Group to the Permutation Group Sn, since the Braid Group also involves switching points around. Indeed there is a surjective homomorphism Bn→Sn however, the Braid Group is infinite while Sn is not. The Braid group will have some extra conditions, In the case of B4

  1. σ1

From Braids to Knots

A link is a collection of knot and they can be formed from taking the closure of a braid. This is done by connecting opposite ends of the strings by lines that do not intersect like such:

Taking the closure of this Braid leads to the figure-8 knot

Alexander's Theorem states that every knot or link is the closure of some Braid. However, there can be many closed Braids that coorespond to the same knot/link. Markov's Theorem gives the conditions that tell whether or not two Braids lead to the same knot/link.

Anyons

So what does this have to do with Physics? We are not so concerned with the exact paths of the particles, but we are concerned with keeping track whether or not some particle moves around the other. One way to encode this information is through a braiding. Aside: This only works in 2 spatial dimensions. If the particles lived in 1 dimensional space exchanging their positions would lead to them passing through each other at somepoint, making it undefined. Furthermore, in 4 dimensional space if one wanted to tie a knot, there would always be that extra spatial degree of freedom availible to untangle it. It is a fact of analysis that knots do not exist in more than 3 dimensions.