UNSPECIFIED (1999) Ordering the braid groups. PACIFIC JOURNAL OF MATHEMATICS, 191 (1). pp. 49-74. ISSN 0030-8730

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Abstract

We give an explicit geometric argument that Artin's braid group Bn 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 Dehornoy (1995) 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 Mosher's (1995) 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Journal or Publication Title:	PACIFIC JOURNAL OF MATHEMATICS
Publisher:	PACIFIC JOURNAL MATHEMATICS
ISSN:	0030-8730
Date:	November 1999
Volume:	191
Number:	1
Number of Pages:	26
Page Range:	pp. 49-74
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/14046

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