Artin braid group

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Definition

A group G is termed an Artin braid group if it can be equipped with a finite presentation as follows. The generators are s_1, s_2, \dots, s_n for some n, and for every i \ne j, we have a nonnegative integer m_{ij}, possibly zero, and such that m_{ij} = m_{ji}. The relation is given by:

s_is_js_i \dots = s_js_i \dots

Both sides have length m_{ij}. When m_{ij} is odd, the left side ends in s_i and the right side ends in s_j. When m_{ij} is even, the left side ends in s_j and the right side ends in s_i.

Such a presentation is termed an Artin presentation or Artin braid presentation.

The standard braid group is an example of an Artin braid group, where m_{ij} = m_{ji} = 3 for all |i - j| = 1 and m_{ij} = m_{ji} = 2 for |i - j| > 1..

Relation with Coxeter groups

Given a group with an Artin presentation, we can consider a corresponding Coxeter group. The Coxeter group is the quotient of the group by the normal closure of the squares of the generators. In other words it is the quotient by the group \langle s_i^2 \rangle . The Coxeter presentation for the Coxeter group obtained has the same m_{ij} as the original group.