Legendrian and transverse cables of positive torus knots

Abstract

In this paper 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 non-destabilizable 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.

34 pages, 6 figures

References in corpus (3)

• An atlas of Legendrian knots
• Studying uniform thickness I: Legendrian simple iterated torus knots
• Studying uniform thickness II: Transversely non-simple iterated torus knots

Cited by in corpus (12)

• An atlas of Legendrian knots
• On transverse invariants from Khovanov homology
• Transverse Surgery on Knots in Contact 3-Manifolds
• Ozsváth-Szabó invariants of contact surgeries
• Contact structures and reducible surgeries
• Rectangular diagrams of surfaces: distinguishing Legendrian knots
• Distinguishing Legendrian knots with trivial orientation-preserving symmetry group
• Legendrian Large Cables And New Phenomenon For Non-Uniformly Thick Knots
• Stein traces and characterizing slopes
• Cabling Legendrian and transverse knots
• A criterion for the Legendrian simplicity of the connected sum
• On The Cost Function Associated With Legendrian Knots