Journal of the European Mathematical Society

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Volume 13, Issue 6, 2011, pp. 1591–1631
DOI: 10.4171/JEMS/289

Published online: 2011-09-16

Every braid admits a short sigma-definite expression

Jean Fromentin[1]

(1) Université de Caen, France

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.

Keywords: Braid group, braid ordering, dual braid monoid, normal form

Fromentin Jean: Every braid admits a short sigma-definite expression. J. Eur. Math. Soc. 13 (2011), 1591-1631. doi: 10.4171/JEMS/289