Homotopy groups of spheres correspond to framed submanifolds of Euclidean space through the Pontjagin-Thom construction.  For example, the Hopf map corresponds to a circle framed “with a twist”.  The homotopy groups of $S^1$ thus correspond to framed codimension one submanifolds.  But such are canonically framed, so there are no interesting/ non-trivial examples.