Studying uniform thickness II: Transversely non-simple iterated torus knots

TL;DR

This paper characterizes when iterated torus knots fail the uniform thickness property, linking it to their cabling structure and transverse non-simplicity, thus advancing understanding of contact structures in knot theory.

Contribution

It provides a complete characterization of iterated torus knots that fail the UTP and connects this to their support of transversely non-simple cabling knot types.

Findings

Iterated torus knots fail UTP iff all iterations are positive cablings.

Knots failing UTP support transversely non-simple cabling types.

The results connect knot cabling properties with contact geometric features.

Abstract

We prove that an iterated torus knot type fails the uniform thickness property (UTP) if and only if all of its iterations are positive cablings, which is precisely when an iterated torus knot type supports the standard contact structure. We also show that all iterated torus knots that fail the UTP support cabling knot types that are transversely non-simple.