Artin Group

In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form

where

.

For, denotes an alternating product of and of length, beginning with . For example,

and

.

If, then there is (by convention) no relation for and .

The integers 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, known as the pure Artin group, is generated by relations of the form .