Geometry & Topology

Braid ordering and the geometry of closed braid

Tetsuya Ito

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

braid group Dehornoy ordering Nielsen–Thurston classification geometric structure


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.

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