Artin braid group

The Artin braid group $𝔅 𝑛$ on $𝑛$ strands is the special case of the more general braid group on the Euclidean plane $ℝ 2$ :

𝔅 𝑛 = 𝔅 𝑛 (ℝ 2) = 𝜋 1 (ℝ 2 𝑛) .

It has presentation $⟨ 𝜎 1, \dots, 𝜎 𝑛 - 1 : 𝑅 ⟩$ topology where $𝑅$ consists of the relations

𝜎 𝑖 𝜎 𝑗 𝜎 𝑖 = 𝜎 𝑗 𝜎 𝑖 𝜎 𝑗 | 𝑖 - 𝑗 | = 1, 𝜎 𝑖 𝜎 𝑗 = 𝜎 𝑗 𝜎 𝑖 | 𝑖 - 𝑗 | \neq 1 .

Proof

proof

The various Artin braid groups may be combined into the Braid category.