Braid ordering and the geometry of closed braid

Ito, Tetsuya

doi:10.2140/gt.2011.15.473

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Braid ordering and the geometry of closed braid

Tetsuya Ito

Geometry & Topology 15 (2011) 473–498

DOI: 10.2140/gt.2011.15.473

Abstract

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.

Keywords

braid group, Dehornoy ordering, Nielsen–Thurston classification, geometric structure

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M50

References

Publication

Received: 30 September 2009

Revised: 14 December 2010

Accepted: 12 December 2010

Published: 25 March 2011

Proposed: Joan Birman

Seconded: David Gabai, Shigeyuki Morita

Authors

Tetsuya Ito
Graduate School of Mathematical Science University of Tokyo 3-8-1 Komaba Meguro-ku 153-8914 Japan
http://www.ms.u-tokyo.ac.jp/~tetitoh/

Volume 15, issue 1 (2011)