Uniqueness of models in persistent homology: the case of curves

We consider generic curves in { {R}}^2 , i.e. generic C¹ functions f:S^1\rightarrow { {R}}^2. We analyze these curves through the persistent homology groups of a filtration induced on S¹ by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to re-parameterizations of S¹. We give a partially positive answer to this question. More precisely, we prove that f = g o h, where h: S¹ → S¹ is a C¹-diffeomorphism, if and only if the persistent homology groups of s o f and s o g coincide, for every s belonging to the group Σ₂ generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s in Σ₂, the persistent Betti number functions of s o f and s o g are close to each other, with respect to a suitable distance.