Legendrian satellites

TL;DR

This paper classifies Legendrian patterns in satellite knots, especially Whitehead and braided patterns, and uses these classifications to understand Legendrian satellite knots and their properties.

Contribution

It provides the first classification of Legendrian Whitehead patterns and advances the understanding of Legendrian satellite knots through pattern classification.

Findings

Legendrian Whitehead patterns classified

Legendrian braided patterns analyzed

New classification results for Legendrian satellite knots

Abstract

In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if the companion knot is Legendrian simple and uniformly thick. This leads to new Legendrian and transverse classification results for knots in the 3-sphere with its standard contact structure as well as a more general perspective on some previous classification results.