On presentations of generalizations of braids with few generators

Abstract

Abstract. In his initial paper on braids E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.The diverse aspects of presentations of braid groups and their generalizations continue to attract attention [12], [5], [15].The canonical presentation of the braid group Br n was given by E. Artin [1] and is well known. It has the generators σ 1 , σ 2 , . . . , σ n−1 and relationsThere exist other presentations of the braid group. J. S. Birman, K. H. Ko and S. J. Lee [5] introduced the presentation with generators a ts with 1 ≤ s < t ≤ n and relations a ts a rq = a rq a ts for (t − r)(t − q)(s − r)(s − q) > 0, a ts a sr = a tr a ts = a sr a tr for 1 ≤ r < s < t ≤ n.The generators a ts are expressed in the canonical generators σ i as follows: a ts = (σ t−1 σ t−2 · · · σ s+1 )σ s (σ −1 s+1 · · · σ −1 t−2 σ −1 t−1 ) for 1 ≤ s < t ≤ n.

About presentations of braid groups and their generalizations

“…We give here the extension of this presentation for the natural generalizations of braids. The results of this section were obtained in [83].…”