Osaka Journal of Mathematics

A new distinguished form for 3-braids

Emille Davie Lawrence

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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

Permanent link to this document
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


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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.