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.Comment: 34 pages, 6 figure

Non-simplicity of Isocontact Embeddings in All Higher Dimensions

“…This continued with [31,Theorem 1.7] where it was shown that the (2, 3)cable of the (2, 3)-torus knot is not transversely simple. Techniques and results have since developed [17,55,56,58] yielding a wide range of applications and classifications results in low-dimensional contact geometry [32,34,59].…”

Naturality of Heegaard Floer invariants under positive rational contact surgery

“…Clearly the small triangles have n w´nz " 0, so we turn to the Chern class term. Referring to (12), the Euler measureχpP S q is defined bŷ…”

On transverse invariants from Khovanov homology

“…It is known that (3) does not necessarily imply (2): by [ELT12], there are transverse knots representing certain cables of torus knots that have the same self-linking number but require an arbitrarily large number of stabilizations to become transversely isotopic. We do not know if (2) necessarily implies (1), although it seems unlikely.…”