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Legendrian and transverse cables of positive torus knots

John B Etnyre, Douglas J LaFountain and Bülent Tosun

Geometry & Topology 16 (2012) 1639–1689

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.

Legendrian, contact, cable, torus knot
Mathematical Subject Classification 2010
Primary: 53D10, 57R17
Secondary: 57M50
Received: 6 May 2011
Revised: 3 March 2012
Accepted: 5 June 2012
Published: 3 August 2012
Proposed: Yasha Eliashberg
Seconded: Peter S Ozsváth, Ronald J Stern
John B Etnyre
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta GA 30332-0160
Douglas J LaFountain
Centre for Quantum Geometry of Moduli Spaces
Aarhus University
Ny Munkegade 118
DK-8000 Aarhus C
Bülent Tosun
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta GA 30332-0160