Uniqueness of models in persistent homology: the case of curves

We consider generic curves in R, i.e. generic C functions f : S → R. We analyze these curves through the persistent homology groups of a filtration induced on S by f . In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to re-parameterizations of S. We give a partially positive answer to this question. More precisely, we prove that f = g ◦ h, where h : S → S is a C-diffeomorphism, if and only if the persistent homology groups of s ◦ f and s ◦ 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 ∈ Σ2, the persistent Betti numbers functions of s◦f and s◦g are close to each other, with respect to a suitable distance. AMS classification scheme numbers: 55N35, 53A04, 68U05