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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

The braid category Definition
The braid group on three strands and its quotient to S₃ Example
The braid group surjects onto the symmetric group Theorem
The two-strand braid group is infinite cyclic Theorem

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)