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.
Citation
Emille Davie Lawrence. "A new distinguished form for 3-braids." Osaka J. Math. 51 (3) 537 - 545, July 2014.
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