Let $(Σ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partialΣ\neq\varnothing$ and punctures $\mathbb{P}\subseteqΣ\setminus\partialΣ$. In this paper we show that for every curve $γ$ on $Σ\setminus\mathbb{P}$, the curve obtained by resolving the kinks of $γ$ in any order is uniquely determined, up to homotopy in $Σ\setminus\mathbb{P}$, by the $2$-orbifold homotopy class of $γ$, in which the punctures are interpreted to be orbifold points of order $2$. Our proof resorts to an application of the Diamond Lemma.