Deformations of curves with constant curvature

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.