Deformations of curves with constant curvature

We prove that curves of constant curvature satisfy the parametric [Formula: see text]-dense relative [Formula: see text]-principle in the space of immersed curves with nonvanishing curvature in Euclidean space [Formula: see text]. It follows that two knots of constant curvature in [Formula: see text] are isotopic, respectively homotopic, through curves of constant curvature if and only if they are isotopic, respectively homotopic, and their self-linking numbers, respectively self-linking numbers mod [Formula: see text], are equal. The proofs are based on convex integration techniques, utilizing parametric versions of classical theorems by Carathéodory and Steinitz in convex geometry.