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Still another approach to the braid ordering

Abstract

We develop a new approach to the linear ordering of the braid group B_nB\_n, based on investigating its restriction to the set \Div(\Delta\_n^d) of all divisors of Δ_nd\Delta\_n^d in the monoid B_∞+B\_\infty^+, i.e., to positive nn-braids whose normal form has length at most dd. In the general case, we compute several numerical parameters attached with the finite orders (\Div(\Delta\_n^d), <). In the case of 3 strands, we moreover give a complete description of the increasing enumeration of (\Div(\Delta\_3^d), <). We deduce a new and specially direct construction of the ordering on B_3B\_3, and a new proof of the result that its restriction to B_3+B\_3^+ is a well-ordering of ordinal type ωω\omega^\omega

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