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

Abstract

We prove that curves of constant torsion satisfy the ${\mathcal{C}}^1$-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.