On the resolution of kinks of curves on punctured surfaces

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