Braid theory

Introduction

Braid theory is an abstract geometric theory in topological mathematics, studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operations correspond to the physical process of concatenating the braids. Braid theory was first introduced by Emil Artin in 1925.

Definition and Basics

In braid theory, the braid group on n strands (denoted B_n), is the group whose elements are equivalence classes of n-braids, and whose group operation is composition of braids. Consider an n-braid as n strands, numbered from 1 to n from top to bottom that are each attached at the top and bottom to a fixed point.

Photograph of a three-strand braid, demonstrating the over-and-under weaving pattern typical of braids.

Braid Groups

Braid groups can be defined algebraically as well as geometrically. The nth braid group B_n is generated by n - 1 elements σ_i subject to the relations:

1. σ_iσ_j = σ_jσ_i for |i - j| > 1 2. σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1 for all i.

Braid Words

A braid can be represented by a braid word, which is a product of generators and their inverses. For example, the braid word σ₁σ₂σ₁^-1 represents a braid on three strands.

Braid Index

The braid index of a braid is the minimum number of strands needed to make it. The braid index of a knot or link is the minimum braid index over all braid representatives of the knot or link.

Braids as Tangles

A tangle is a proper embedding of a 1-dimensional manifold with boundary in the 3-ball. A braid can be considered as a tangle whose ends are attached to the boundary of the 3-ball in a particular way.

Braids in Physics and Chemistry

Braid theory has been applied in several areas of physics and chemistry, including quantum mechanics, fluid dynamics, and DNA sequencing.

Braids in Computer Science

In computer science, braid theory is used in the study of algorithms, particularly those involving sorting and data manipulation.