Arnold-type Invariants of Curves on Surfaces

Recently V. Arnold introduced Strangeness and $J^{\pm }$ invariants of generic immersions of an oriented circle to ${\mathbb{R}}^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by V. Arnold.