On the resolution of kinks of curves on punctured surfaces

Abstract

Let $(\Sigma ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partial \Sigma \ne \varnothing$ 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.

v1: 23 pages, 18 figures; v2: improvements following comments by the referee, one figure added, resolution of figures improved, 24 pages, 19 figures, to appear in Algebraic & Geometric Topology

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