Topology of the space of nondegenerate curves

A curve on a sphere or on a projective space is called nondegenerate if it has a nondegenerate moving frame at every point. The number of homotopy classes of closed nondegenerate curves immersed in the sphere or projective space is computed. In the case of the sphere $S^n$, this turns out to be 4 for odd $n\geqslant 3$ and 6 for even $n\geqslant 2$; in the case of the projective space $\mathbf P^n$, 10 for odd $n\geqslant 3$ and 3 for even $n\geqslant 2$.