Braid Groups

The braid group encodes the ways strands can be intertwined, giving an algebraic structure whose closures produce every knot and link and whose representations yield knot invariants.

Definition

The braid group on n strands is the group with generators that interchange adjacent strands, subject to the braid relations; it is simultaneously the fundamental group of the configuration space of n points in the plane and the mapping class group of the n-punctured disk.

Scope

This topic introduces the Artin braid group via generators and relations, its description as the fundamental group of a configuration space and as a mapping class group of the punctured disk, and the word and conjugacy problems solved through Garside normal forms. It develops the link between braids and links through Alexander's theorem (every link is a braid closure) and Markov's theorem (which braids close to the same link), and representations such as the Burau and the Temperley-Lieb representation that give rise to the Jones polynomial.

Core questions

• What relations define the braid group, and why do they capture intertwining of strands?
• How does Alexander's theorem realize every link as the closure of a braid?
• Which braids close to the same link, as answered by Markov's theorem?
• How do representations of the braid group produce knot invariants like the Jones polynomial?

Key concepts

• Artin generators and the braid relations
• Braid group as configuration space and mapping class group
• Alexander's and Markov's theorems linking braids and links
• Garside normal form and the word problem
• Burau and Temperley-Lieb representations

Clinical relevance

Braid groups are central to the construction of quantum knot invariants, to the theory of mapping class groups and surface topology, and to topological quantum computation, where braiding of anyons realizes quantum gates.

History

Artin defined and studied the braid group in his 1925 and 1947 papers, establishing the generators, relations, and the word problem; Markov's theorem and the later representation-theoretic constructions tied braids to knot invariants and, through Jones, to operator algebras.

Key figures

• Emil Artin
• Andrey Markov Jr.
• Vladimir Turaev

Seminal works

• kassel2008
• artin1947

Frequently asked questions

How are braids related to knots?

Closing up a braid by joining each strand's top to its bottom produces a knot or link; Alexander's theorem says every link arises this way, and Markov's theorem describes exactly when two braids yield the same link.

Why are braid groups relevant to quantum computing?

In topological quantum computation, quantum information is stored in anyons and processed by braiding them; the braid group governs these operations, making its representations a model for fault-tolerant quantum gates.