Cabling Legendrian and transverse knots

Abstract

In this paper we will show how to classify Legendrian and transverse knots in the knot type of “sufficiently positive” cables of a knot in terms of the classification of the underlying knot. We will also completely explain the phenomena of “Legendrian large” cables. These are Legendrian representatives of cables that have Thurston–Bennequin invariant larger than the framing coming from the cabling torus. Such examples have only recently, and unexpectedly, been found. We will also give criteria that determines the classification of Legendrian and transverse knots the knot type of negative cables.