Horizontal loops in Engel space

TL;DR

This paper provides a straightforward proof classifying embedded circles tangent to the standard Engel structure in Euclidean 4-space, showing they are distinguished by their rotation number.

Contribution

It offers a simple proof of the classification of tangent embedded circles in Engel space based on their rotation number, confirming a result first observed by J. Adachi.

Findings

Embedded circles tangent to Engel structure are classified by rotation number.

The proof simplifies the understanding of tangent circle embeddings in Engel space.

Classification is up to isotopy within the Engel structure.

Abstract

A simple proof is given of the following result first observed by J. Adachi: embedded circles tangent to the standard Engel structure on Euclidean 4-space are classified, up to isotopy via such embeddings, by their rotation number.