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Studying uniform thickness I: Legendrian simple iterated torus knots

Douglas J LaFountain

Algebraic & Geometric Topology 10 (2010) 891–916

We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian nonsimplicity in the class of iterated torus knots, including specific conditions on knot types.

Legendrian knots, convex surfaces, uniform thickness property
Mathematical Subject Classification 2000
Primary: 57M25, 57R17
Secondary: 57M50
Received: 12 June 2009
Revised: 17 December 2009
Accepted: 4 March 2010
Published: 13 April 2010
Douglas J LaFountain
Department of Mathematics
University at Buffalo
244 Mathematics Building
Buffalo, NY 14260