Classification of tangent and transverse knots in bracket-generating distributions

TL;DR

This paper establishes comprehensive h-principles for classifying embedded horizontal and transverse curves in high-dimensional manifolds with bracket-generating distributions, highlighting differences from the 3D contact case.

Contribution

It proves complete h-principles for embedded regular horizontal and transverse curves in manifolds of dimension greater than 3, extending understanding beyond the 3D contact scenario.

Findings

Complete h-principles for embedded regular horizontal curves

Complete h-principles for embedded transverse curves

Analogous results for immersions without ambient dimension restrictions

Abstract

Consider a manifold, of dimension greater than 3, equipped with a bracket-generating distribution. In this article we prove complete h-principles for embedded regular horizontal curves and for embedded transverse curves. These results contrast with the 3-dimensional contact case, where the full h-principle for transverse/legendrian knots is known not to hold. We also prove analogous statements for immersions, with no assumptions on the ambient dimension.