Proposition (C) (comparison): Every braid

The linear ordering of braids, which was first discovered using results of self-distributive al-gebra, has now received several alternative constructions. Here, we mention some of them, in particular the recent approach de-velopped by Ivan Dynnikov using laminations. 1 The linear ordering of braids • Standard presentation of Artin’s braid group Bn:〈 σ1,..., σn−1 σiσjσi = σiσjσi for |i − j | = 1σiσj = σjσi for |i − j | ≥ 2 with geometric interpretation: 1 2 i i+1 n σi:...... Proposition (A) (acyclicity): A σ1-positive braid is not trivial. • σ1-positive = admits at least one expression where σ1 appears but σ −1 1 does no