Studying uniform thickness I: Legendrian simple iterated torus knots

TL;DR

This paper proves that certain classes of iterated torus knots are Legendrian simple and satisfy the uniform thickness property, especially those starting with negative torus knots, and explores conditions for non-simplicity.

Contribution

It establishes the closure of Legendrian simplicity and UTP under cabling for specific knot classes and identifies new conditions for non-simplicity in iterated torus knots.

Findings

All iterated cabling knot types starting with negative torus knots are Legendrian simple.

Many iterated cablings starting with positive torus knots are Legendrian simple and satisfy UTP.

New necessary conditions for failure of UTP and Legendrian non-simplicity in iterated torus knots.

Abstract

We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian non-simplicity in the class of iterated torus knots, including specific conditions on knot types.