Still another approach to the braid ordering

Patrick Dehornoy

Vol. 232 (2007), No. 1, 139–175

DOI: 10.2140/pjm.2007.232.139

Abstract

We develop a new approach to the linear ordering of the braid group B_n, based on investigating its restriction to the set Div(Δ_n^d) of all divisors of Δ_n^d in the monoid B_∞⁺, that is, to positive n-braids whose normal form has length at most d. In the general case, we compute several numerical parameters attached with the finite orders Div(Δ_n^d). In the case of 3 strands, we moreover give a complete description of the increasing enumeration of Div(Δ₃^d). We deduce a new and especially direct construction of the ordering on B₃, and a new proof of the result that its restriction to B₃⁺ is a well-ordering of ordinal type ω^ω.

Keywords

braid group, orderable group, well-ordering, normal form, fundamental braid

Mathematical Subject Classification 2000

Primary: 20F36

Secondary: 05A05, 20F60

Milestones

Received: 6 April 2006

Accepted: 24 May 2006

Published: 1 September 2007