Geodesic
Every braid admits a short sigma-definite expression
Journal of the European Mathematical Society, Tome 13 (2011) no. 6, pp. 1591-1631 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/jems/289
Classification : 20-XX, 06-XX, 00-XX
Keywords: Braid group, braid ordering, dual braid monoid, normal form 
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     author = {Jean Fromentin},
     title = {Every braid admits a short sigma-definite expression},
     journal = {Journal of the European Mathematical Society},
     pages = {1591--1631},
     year = {2011},
     volume = {13},
     number = {6},
     doi = {10.4171/jems/289},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/289/}
}
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Jean Fromentin. Every braid admits a short sigma-definite expression. Journal of the European Mathematical Society, Tome 13 (2011) no. 6, pp. 1591-1631. doi: 10.4171/jems/289

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