What are Braids?

We think of a braid as a set of vertical strings connecting two horizontal parallel bars in 3 dimensional space. These strings may cross each other at various points but may NOT travel back upon themselves or cross the bars.

Braids are not themselves knots, but can be used as representatives of knots and links. Suppose that we stretched the strings to allow us to bring the bars around and connect them to one another. The resulting loop (called the closure of the braid) forms a knot or a link, and all knots and links have closed braid representations. A closed braid representation can be found by arranging the strands of a knot or link to “travel” clockwise around a pre-determined “axis” point, without backtracking.

Braids can be described as sequences of symbols known as Braid Words. In a braid word, we write to denote the -th strand of the braid crossing over the strand, and to denote the -th strand of the braid crossing under the strand. We write these symbols in the order we encounter their respective crossings as we travel from the top of the braid to the bottom. Note that if we have next to a Redemeister II move removes the pair without changing the braid.

This cancellation is the first of a set of “moves” we can use to rearrange the braid without changing its properties – similar to the Redemeister moves in knots. If one braid can be made into another by a sequence of these moves, then the braids must be equivalent. The moves are as follows:
1)Adding or removing or

2) if the difference between and is greater than or equal to 2.

3) if the difference between i and j is exactly 1. Under the same restriction .

There exists a multiplication on braids, involving attaching one -strand braid underneath another -strand braid. Braids of -strand braids form a group under this Braid Multiplication.