Classification of Engel Knots

TL;DR

This paper proves a homotopy equivalence between the space of Engel knots and formal Engel knots in certain conditions, establishing a close h-principle for Engel knot classification.

Contribution

It establishes a relative, parametric, and close h-principle for Engel knots, showing the scanning map is a weak homotopy equivalence under specified conditions.

Findings

Scanning map is a weak homotopy equivalence for Engel knots

Establishes a relative, parametric, and close h-principle

Provides new insights into the topology of Engel knots

Abstract

For any Engel 4-fold, we show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the orbits of the Engel kernel. This is a relative, parametric and close h-principle.