Deformations of curves with constant curvature

TL;DR

This paper proves that curves with constant curvature in Euclidean space satisfy a dense relative h-principle, leading to classification results for knots of constant curvature based on isotopy, homotopy, and self-linking invariants.

Contribution

It establishes a parametric $C^1$-dense relative $h$-principle for curves of constant curvature in Euclidean space, connecting geometric properties with topological classifications.

Findings

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

Knots of constant curvature in $R^3$ are classified by isotopy, homotopy, and self-linking numbers.

Two knots of constant curvature are isotopic or homotopic iff their invariants match.

Abstract

We prove that curves of constant curvature satisfy the parametric $C^1$-dense relative $h$-principle in the space of immersed curves with nonvanishing curvature in Euclidean space $R^{n\geq 3}$. It follows that two knots of constant curvature in $R^3$ are isotopic, resp. homotopic, through curves of constant curvature if and only if they are isotopic, resp. homotopic, and their self-linking numbers, resp. self-linking numbers mod $2$, are equal.