Still another approach to the braid ordering

2005/06/24 by Patrick Dehornoy, Dehornoy, Patrick
Mathematics · #05A05 #20F36 #20F60 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:05A05 #msc:20F36 #msc:20F60

paper · pdf · doi:10.48550/arxiv.math/0506495

arxiv created 2005/06/24 · arxiv updated 2009/12/01

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_∞⁺, i.e., 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(Δ_3^d), <). We deduce a new and specially direct construction of the ordering on B_3, and a new proof of the result that its restriction to B_3⁺ is a well-ordering of ordinal type ω^ω.

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