%X 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≤p<21≤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 SLEκSLEκ null set, where 0<κ≤40<κ≤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. %O This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. %L discovery1555643 %K Rough path theory; Uniqueness of signature problem; SLE curves %J Journal of Functional Analysis %P 1778-1806 %D 2014 %A H Boedihardjo %A H Ni %A Z Qian %V 267 %T Uniqueness of signature for simple curves %N 6