Volume 15, issue 1 (2011)

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Braid ordering and the geometry of closed braid

Tetsuya Ito

Geometry & Topology 15 (2011) 473–498

We study the relationships between the Dehornoy ordering of the braid groups and the topology and geometry of the closed braid complements. We show that the Dehornoy floor of braids, which is a nonnegative integer determined by the Dehornoy ordering, tells us the position of essential surfaces in the closed braid complements. Furthermore, we prove that if the Dehornoy floor of a braid is bigger than or equal to two, then the Nielsen–Thurston classification of braids and the geometric structure of the closed braid complements are in one-to-one correspondence.

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braid group, Dehornoy ordering, Nielsen–Thurston classification, geometric structure
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M50
Received: 30 September 2009
Revised: 14 December 2010
Accepted: 12 December 2010
Published: 25 March 2011
Proposed: Joan Birman
Seconded: David Gabai, Shigeyuki Morita
Tetsuya Ito
Graduate School of Mathematical Science
University of Tokyo
3-8-1 Komaba
Meguro-ku 153-8914