Geometry & Topology

Braid ordering and the geometry of closed braid

Tetsuya Ito

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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.

Article information

Source
Geom. Topol., Volume 15, Number 1 (2011), 473-498.

Dates
Received: 30 September 2009
Revised: 14 December 2010
Accepted: 12 December 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732283

Digital Object Identifier
doi:10.2140/gt.2011.15.473

Mathematical Reviews number (MathSciNet)
MR2788641

Zentralblatt MATH identifier
1214.57010

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
braid group Dehornoy ordering Nielsen–Thurston classification geometric structure

Citation

Ito, Tetsuya. Braid ordering and the geometry of closed braid. Geom. Topol. 15 (2011), no. 1, 473--498. doi:10.2140/gt.2011.15.473. https://projecteuclid.org/euclid.gt/1513732283


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