On the Legendrian and transverse classification of cablings

TL;DR

This paper investigates the classification of Legendrian and transverse knots within (p,q)-cable knot types, providing structural theorems that determine when these cable knots are also Legendrian simple.

Contribution

It introduces two structural theorems that characterize when the (p,q)-cable of a Legendrian simple knot type remains Legendrian simple.

Findings

Identifies conditions for Legendrian simplicity of cable knots

Provides structural theorems for classification

Advances understanding of Legendrian and transverse knot types

Abstract

In this note we study Legendrian and transverse knots in the knot type of a (p,q)-cable of a knot K in 3-sphere. We give two structural theorems that describe when the (p,q)-cable of a Legendrian simple knot type K is also Legendrian simple.