曲线图中的折纸边路径
Origami edge-paths in the curve graph
摘要：
通过将每个右边缘粘到左边缘和每个上边缘粘到下边缘，从单位欧几里德正方形的有限集合获得属$g \geq 2$的闭合定向表面$S_g$上的“折纸”（或平坦结构）。本文的主要研究对象是“折纸曲线对”--在$S_g$中填充简单闭曲线对$ (α,β)$，使它们的最小交等于它们的代数交--它们是“相干的”。一对折纸曲线自然地与$S_g$上的折纸相关联。我们的主要结果建立，对于任何折纸对曲线存在一个“折纸边路径”，一系列的曲线，$α=α_0, α_1, α_2, \cdots, α_n = β$，这样：$α_i$相交$α_{i+1}$恰好一次;任何对$(α_i, α_j)$是连贯的;因此，任何填充对，$(α_i, α_j)$，也是折纸。随着它们的存在建立，我们提供最短的折纸边缘路径作为一个领域的调查。
英文摘要：
An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main objects of study in this note are "origami pairs of curves" -- filling pairs of simple closed curves, $ (α,β)$, in $S_g$ such that their minimal intersection is equal to their algebraic intersection -- they are "coherent". An origami pair of curves is naturally associated with an origami on $S_g$. Our main result establishes that for any origami pair of curves there exists an "origami edge-path", a sequence of curves, $α=α_0, α_1, α_2, \cdots, α_n = β$, such that: $α_i$ intersects $α_{i+1}$ at exactly once; any pair $(α_i, α_j)$ is coherent; and thus, any filling pair, $(α_i, α_j)$, is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
