Geometry & Topology

Legendrian and transverse cables of positive torus knots

John B Etnyre, Douglas J LaFountain, and Bülent Tosun

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We classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have nondestabilizable Legendrian representatives whose Thurston–Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study “partially thickenable” tori, which allow us to completely classify solid tori representing positive torus knots.

Article information

Geom. Topol., Volume 16, Number 3 (2012), 1639-1689.

Received: 6 May 2011
Revised: 3 March 2012
Accepted: 5 June 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D10: Contact manifolds, general 57R17: Symplectic and contact topology
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Legendrian contact cable torus knot


Etnyre, John B; LaFountain, Douglas J; Tosun, Bülent. Legendrian and transverse cables of positive torus knots. Geom. Topol. 16 (2012), no. 3, 1639--1689. doi:10.2140/gt.2012.16.1639.

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