Deformations of curves with constant curvature

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 $R^{n\ge 3}$. 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. The proofs are based on convex integration techniques, utilizing parametric versions of classical theorems by Carathéodory and Steinitz in convex geometry.

arXiv comment23 pages, 2 figures; Intermediate results strengthened to close a gap; Accepted for publication in J. Topol. Anal