Still another approach to the braid ordering

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 $\Delta \_n^d$ in the monoid $B\_{\infty }^{+}$, 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 ${\omega }^{\omega }$.