On the resolution of kinks of curves on punctured surfaces

P / be a surface with marked points M @ and punctures P n @ .We show that for every curve on n P , the curve obtained by resolving the kinks of in any order is uniquely determined, up to homotopy in n P , by the 2-orbifold homotopy class of , in which the punctures are interpreted to be orbifold points of order 2. Our proof relies on an application of the diamond lemma.