The Algebraical Braid Group

In his paper 'Theorie der Z6pfe' E. Artin' presented a theory of braids based on a study of their projections on a two-dimensional plane. In the projection each strand of a braid appears as a line, vertical in general, but at certain levels two neighboring strands interchange position, one strand crossing in front of the other one. The crossing of the strand in position i in front of the strand in position (i + 1) is denoted by as . If the latter crosses in front of the former the crossing is denoted by ai A braid is completely described as a power product in the a's. The braids aiaj and ajai for I i j I > 2 are obviously geometrically equivalent and the same holds for the braids aiai+iai and ai+1iOiaO . These equivalences are defining relations for the braid group. In other words, this group is derived from the free group generated by the a's by introducing the relations aiaj / ajai ( I i-j i > 2) and aiaiojai ai+1aiai . These results were fully recognized in the paper of Artin mentioned above and led in particular to a classification of braids. The proofs, however, are partially intuitive. In a recent paper2 Artin returned to this question making use of a more direct approach which permits an elegant and completely rigorous treatment of the problem. Instead of utilizing the projection, braid coordinates are introduced. The a's appear at a late stage in the theory but even then they play only a minor role. By its nature the problem is geometrical and throughout the major part of his paper Artin makes extensive use of geometrical considerations. In the present paper a purely group-theoretical problem is considered. The 'algebraical braid group' is defined as the group generated by the a's with the relations mentioned above. This group is analyzed by algebraical methods and this analysis leads to the same fundamental results obtained by Artin; first a criterion for the equality of two elements of the group, secondly a true representation of the group as a group of substitutions in a free group. By combining the geometrical theory with the algebraical approach it is finally shown that the defining relations in the a's are defining relations for the geometrical braid group.