Uniqueness of signature for simple curves

Abstract

We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1\leq p<2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLE_{\kappa} null set, where 0<\kappa\leq 4, the signature series of curves uniquely determine the curves. Our calculations also enable us to express the Fourier transform of the n-point functions of SLE curves in terms of the expected signature of SLE curves. Although the techniques used in this article are deterministic, the results provide a platform for studying SLE curves through the signatures of their sample paths.