Homotopy index polynomials for knotoids

We define homotopy index polynomials and a homotopy type of knotoids in an oriented surface, and consider some basic properties of these polynomials. We show that for planar knotoids the homotopy polynomials are not weaker than previously defined index polynomials, and that the homotopy index polynomials detect non-rotatability of spherical knotoids.