Artin group
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In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form
where
.
For 
, 
represents an alternating product of xi and xj of length m, beginning with xi. (For example, 
and 
.) If 
, then there is no relation for xi and xj.
The mi,j can be organized into a symmetric matrix, known as the Coxeter matrix of the group. Each Artin group has as a quotient the Coxeter group with the same set of generators and Coxeter matrix. The kernel of the homomorphism to the associated Coxeter group is generated by relations of the form 
.
Braid groups are examples of Artin groups, with Coxeter matrix mi,i + 1 = 3 and mi,j = 2 for | i − j | > 1.
