Braids, the Artin Group, and the Jones Polynomial

This paper is about Braids and the Artin braid group Bn. After some initial definitions and examples, I proceed to show how the Jones polynomial can be derived through a representation of the braid group by the Temperley-Lieb Algebra, an approach similar to Jones’ original construction. 1 Braids, An Introduction Perhaps the most obvious place to start is with the definition [3] of a braid: Definition 1 Consider two parallel planes A and B in R, each containing n distinct points {ai} and {bi} respectively. Then an n-strand braid is a collection of n curves {xi} such that: 1. Each xi has one endpoint at an ai and an endpoint at a bi. 2. All the xi are pairwise disjoint. 3. Every plane parallel to A and B intersects each of the xi at one point or not at all.