Every braid admits a short sigma-definite expression

A result by Dehornoy (1992) says that every nontrivial braid admits a σ -definite expression, defined as a braid word in which the generator σi with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a σ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.