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Ordering the braid groups. (English) Zbl 1009.20042
Summary: We give an explicit geometric argument that Artin’s braid group \(B_n\) is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call ‘left-consistent canonical form’. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of P. Dehornoy [J. Knot Theory Ramifications 4, No. 1, 33-79 (1995; Zbl 0873.20030)] constructed by very different means, and we recover Dehornoy’s main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both.
Our definition of order is strongly connected with L. Mosher’s [Ann. Math. (2) 142, No. 2, 303-384 (1995; Zbl 0867.57004)] normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word.

20F36 Braid groups; Artin groups
57M07 Topological methods in group theory
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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