On the resolution of kinks of curves on punctured surfaces

TL;DR

This paper proves that resolving kinks in curves on punctured surfaces yields a unique homotopy class determined by an orbifold interpretation, using the Diamond Lemma for the proof.

Contribution

It establishes the uniqueness of kink resolution for curves on punctured surfaces within an orbifold framework, extending prior understanding of curve homotopies.

Findings

Kink resolution is unique up to homotopy in punctured surfaces.

The orbifold perspective with order 2 points is key to the proof.

Application of the Diamond Lemma ensures the resolution process is well-defined.

Abstract

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.