$h$-Principles for curves and knots of constant torsion

TL;DR

This paper proves that curves with constant torsion are flexible in the $C^1$ topology, allowing for dense approximation and the existence of knots in each isotopy class, using convex integration and degree theory.

Contribution

It establishes the $C^1$-density of the h-principle for constant torsion curves and demonstrates the existence of knots within each isotopy class.

Findings

Curves of constant torsion satisfy the $C^1$-dense h-principle.

Existence of knots of constant torsion in every isotopy class.

Methods involve convex integration and degree theory.

Abstract

We prove that curves of constant torsion satisfy the $C^1$-dense h-principle in the space of immersed curves in Euclidean space. In particular, there exists a knot of constant torsion in each isotopy class. Our methods, which involve convex integration and degree theory, quickly establish these results for curves of constant curvature as well.