The Braid Order

Abstract

In this chapter, we construct a linear ordering of the braids. This linear ordering appears canonical in several respects; in particular, the braids larger than 1 are characterized as those braids admitting an expression where the generator with smaller index appears positively onlyi.e.where some letter σ _i occurs, but the letter σ ⁻¹ _i does not, nor does any letter σ ^± _k with k <i. The order is decidablei.e.there exists an effective algorithm that compares any two given braid words, it is compatible with multiplication on one side, and the setB _∞ of all braids is order isomorphic to the rationals.