TY - JOUR Y1 - 2014/09/15/ UR - https://doi.org/10.1016/j.jfa.2014.06.006 ID - discovery1555643 N2 - 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 SLE?SLE? null set, where 0<??40<??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. SP - 1778 VL - 267 JF - Journal of Functional Analysis A1 - Boedihardjo, H A1 - Ni, H A1 - Qian, Z AV - public TI - Uniqueness of signature for simple curves SN - 0022-1236 EP - 1806 IS - 6 N1 - This version is the author accepted manuscript. For information on re-use, please refer to the publisher?s terms and conditions. KW - Rough path theory; Uniqueness of signature problem; SLE curves ER -