## Osaka Journal of Mathematics

### A new distinguished form for 3-braids

Emille Davie Lawrence

#### Abstract

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

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

Dates
First available in Project Euclid: 23 October 2014

https://projecteuclid.org/euclid.ojm/1414090789

Mathematical Reviews number (MathSciNet)
MR3272603

Zentralblatt MATH identifier
1304.62147

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

#### Citation

Lawrence, Emille Davie. A new distinguished form for 3-braids. Osaka J. Math. 51 (2014), no. 3, 537--545. https://projecteuclid.org/euclid.ojm/1414090789

#### References

• 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.