On Presentations of Generalizations of Braids with Few Generators

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 Brn was given by E. Artin [1] and is well known. It has the generators σ1, σ2, . . . , σn−1 and relations { σiσj = σj σi, if |i− j| > 1, i, j = 1, ..., n− 1; σiσi+1σi = σi+1σiσi+1, i = 1, ..., n− 2. There exist other presentations of the braid group. J. S. Birman, K. H. Ko and S. J. Lee [5] introduced the presentation with generators ats with 1 ≤ s < t ≤ n and relations { atsarq = arqats for (t− r)(t− q)(s− r)(s− q) > 0, atsasr = atrats = asratr for 1 ≤ r < s < t ≤ n. The generators ats are expressed in the canonical generators σi as follows: ats = (σt−1σt−2 · · ·σs+1)σs(σ −1 s+1 · · ·σ −1 t−2σ −1 t−1) for 1 ≤ s < t ≤ n. An analogue of the Birman-Ko-Lee presentation for the generalizations of braids (namely, for the singular braid monoid) was obtained in [15]. In the initial paper [1] Artin gave another presentation of the braid group, with two generators, say σ1 and σ, and the following relations: (1) { σ1σ σ1σ −i = σσ1σ σ1 for 2 ≤ i ≤ n/2, σ = (σσ1) . The connection with the canonical generators is given by the formulae: (2) σ = σ1σ2 . . . σn−1, (3) σi+1 = σ σ1σ , i = 1, . . . n− 2. This presentation was also discussed in the book by H. S. M. Coxeter and W. O. J. Moser [8]. It is interesting to obtain the analogues of the presentations of the type (1) for various generalizations of braids [6], [9], [2], [4], [11], [14]. 2000 Mathematics Subject Classification. Primary 20F36; Secondary 20F38, 20M05.