DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The braid group by Artin presentation

Definition

For $n \geq 2$ , the braid group $B_{n}$ is the group with generators $σ_{1}, \dots, σ_{n - 1}$ and relations

$σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1} (1 \leq i \leq n - 2),$

$σ_{i} σ_{j} = σ_{j} σ_{i} (1 \leq i, j \leq n - 1, ∣ i - j ∣ > 1) .$

Equivalently,

$B_{n} = ⟨ σ_{1}, \dots, σ_{n - 1} ∣ σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1} (1 \leq i \leq n - 2), σ_{i} σ_{j} = σ_{j} σ_{i} (1 \leq i, j \leq n - 1, ∣ i - j ∣ > 1) ⟩,$

interpreted in the sense of Group presentation by generators and relations and Relators and relations; finitely generated, finitely related, and finite presentations.

For $n = 0$ and $n = 1$ , there are no generators, and $B_{n}$ is the trivial group given by the empty presentation.

Depends on

Used by

All four classical braid models realize the Artin presentation Corollary
An object of a braided category carries canonical braid actions Corollary
The braid group word problem is decidable by garside normal form Corollary
A linear representation need not be faithful Counterexample
Algebraic braid relations do not imply unitarity Counterexample
Conjugacy alone does not classify braid closures Counterexample
Exponent sum is not a complete braid normal form Counterexample
The induced permutation does not determine a braid Counterexample
Braid action on a fixed-charge fusion space Definition
Exponent sum and writhe of a braid Definition
Markov conjugation and stabilization moves Definition
Positive braid monoid Definition
The Artin representation on a free group Definition
The braid category Definition
The coloured reduced Burau matrix Definition
The exponent sum of a braid Definition
The Khovanov-Rozansky complex and trigraded braid homology Definition
The normalized Khovanov-Rozansky HOMFLYPT Euler series Definition
The ribbon evaluation of an X-colored closed braid Definition
The total winding homomorphism of the punctured disk Definition
The two-variable covering homomorphism Definition
The unreduced Burau matrices Definition
The Zariski combing words alphaᵢ and xᵢ in the Artin presentation Definition
A left garside normal form computation in b three Example
Combing a four-strand braid word Example
Normalized comparison maps around a relation loop Example
The braid group on three strands and its quotient to S₃ Example
The Burau determinant for a two-strand torus link Example
The free-kernel words for three-strand braid combing Example
The full twist in b three Example
The Yamada-Vogel algorithm on a small diagram Example
Torus links as closures of two-strand braids Example
Worked Fibonacci and Ising braid matrices Example
A central positive braid is a power of delta squared for n greater than two Lemma
A height-zero diagram represents a closed braid Lemma
A nontrivial five-strand braid lies in the Burau kernel Lemma
Band exchanges decompose into ordinary Markov moves Lemma
Block interchanges transport arbitrary braid boxes Lemma
Braid-isotopic closed braids are conjugate Lemma
Braid-like Reidemeister moves on closed braids are braid isotopies Lemma

…and 44 more results.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michael Muger, Tensor Categories: A Selective Guided Tour, Section 4 (standard reference, not scraped)