Uniqueness of models in persistent homology: the case of curves

We consider generic curves in R^2, i.e. generic C^1 functions f : S^1 → R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f . In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to reparameterizationsof S^1. We give a partially positive answer to this question.More precisely, we prove that f = g ◦ h, where h : S^1 → S^1 is a C^1-diffeomorphism, if and only if the persistent homology groups of s ◦ f and s ◦g coincide, for every s belonging to the group S 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 themax-norm (up to re-parameterizations)if and only if, for every s in S, the persistent Betti number functions of s ◦ f and s ◦ g are close to each other, with respect to a suitable distance.