On the resolution of kinks of curves on punctured surfaces

TL;DR

The paper addresses canonical normalization of curves on punctured surfaces by treating punctures as $2$-order orbifold points and analyzing kink resolutions. It develops a combinatorial framework using the leafy dual graph of a signature-zero triangulation and encodes curves as backtrack-free walks, then applies the Diamond Lemma to show that resolving kinks yields a unique kink-free representative within the $2$-orbifold class, independent of triangulation. By constructing quotient groupoids that compare orbifold and ordinary fundamental groupoids via strong deformation retractions, the authors establish explicit isomorphisms and a canonical map $\iota$ assigning kink-free representatives. The results provide a robust, triangulation-independent procedure for passing from orbifold to ordinary curve classes, with implications for cluster theory and representation theory where punctures are modeled as orbifold points. Overall, the work clarifies when kink elimination preserves the essential homotopy data and furnishes canonical representatives for both orbifold and non-orbifold contexts.

Abstract

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.

Figures (17)

• Figure 1:
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 7:

Theorems & Definitions (44)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Remark 2.5
• Theorem 2.6
• Definition 2.7
• Example 2.8
• Definition 3.1