The Artin representation on a free group

By The Artin automorphisms satisfy the braid relations the assignment ${\sigma }_i\mapsto \rho ({\sigma }_i)$ satisfies the defining relations of the presented braid group $B_n$ of The braid group by Artin presentation, so von Dyck's theorem Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group yields a unique homomorphism $\rho :B_n⟶\mathrm{Aut}(F_n)\mathrm{.}$ This is the Artin representation; it is the frozen convention for every later statement of the page.

Uniqueness and effectivity. The homomorphism is unique because it is prescribed on the generating set $\{{\sigma }_1,\dots ,{\sigma }_{n-1}\}$, and it is computed on a braid word by composing the finitely many automorphisms $\rho ({\sigma }_i{)}^{\pm 1}$ attached to its letters, as frozen in Artin automorphisms of the free group. No choice principle and no geometric input are used in the construction; von Dyck's theorem Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group supplies existence and uniqueness of the extension.

• For $n\le 1$ the group $B_n$ is trivial, so $\rho$ is the unique homomorphism from the trivial group and the assertion is vacuous.
• The construction uses the abstract presentation only; that the abstract group is the mapping class group of the punctured disk, and that its generator acts by the frozen Nielsen substitutions, is proved separately in prop-the-geometric-action-on-meridians-is-the-artin-representation.