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We consider generic curves in R^2, i.e. generic C^1 functions f from S^1 toR^2. We analyze these curves through the persistent homology groups of afiltration induced on S^1 by f. In particular, we consider the question whetherthese persistent homology groups uniquely characterize f, at least up tore-parameterizations of S^1. We give a partially positive answer to thisquestion. More precisely, we prove that f=goh, where h:S^1-> S^1 is aC^1-diffeomorphism, if and only if the persistent homology groups of sof andsog coincide, for every s belonging to the group Sigma_2 generated byreflections in the coordinate axes. Moreover, for a smaller set of genericfunctions, we show that f and g are close to each other in the max-norm (up tore-parameterizations) if and only if, for every s belonging to Sigma_2, thepersistent Betti numbers functions of sof and sog are close to each other, withrespect to a suitable distance.

Uniqueness of models in persistent homology: the case of curves