Osaka Journal of Mathematics

A new distinguished form for 3-braids

Emille Davie Lawrence

Full-text: Open access


We show that every $3$-strand braid has a representative word of a given form, and furthermore, this form allows us, in most cases, to deduce positivity (or negativity) in the $\sigma$-ordering of $B_{3}$. The $\sigma$-ordering of $B_{n}$ was introduced by Patrick Dehornoy in the late 1990's, however, other (equivalent) orderings were discovered soon after by Fenn, Greene, Rolfsen, et al.

Article information

Osaka J. Math., Volume 51, Number 3 (2014), 537-545.

First available in Project Euclid: 23 October 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M07: Topological methods in group theory
Secondary: 20F60: Ordered groups [See mainly 06F15]


Lawrence, Emille Davie. A new distinguished form for 3-braids. Osaka J. Math. 51 (2014), no. 3, 537--545.

Export citation


  • E. Artin: Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72.
  • P. Dehornoy: A fast method for comparing braids, Adv. Math. 125 (1997), 200–235.
  • P. Dehornoy: Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), 115–150.
  • P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest: Ordering Braids, Mathematical Surveys and Monographs 148, Amer. Math. Soc., Providence, RI, 2008.
  • R. Fenn, M.T. Greene, D. Rolfsen, C. Rourke and B. Wiest: Ordering the braid groups, Pacific J. Math. 191 (1999), 49–74.
  • F.A. Garside: The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235–254.