论文标题
曲线图中的折纸边缘路径
Origami edge-paths in the curve graph
论文作者
论文摘要
从有限的单元欧几里得正方形集合中获得的封闭式表面上的“折纸”（或平坦的结构），$ g \ geq 2 $是通过将每个右边缘粘合到左一个顶部边缘到底部的有限单位欧几里得正方形的。本说明中的研究主要对象是“折纸对曲线” - 填充一对简单的封闭曲线，$（α，β）$，$ s_g $，使它们的最小交叉点等于其代数交叉点 - 它们是“相干”。折纸对曲线自然与$ s_g $上的折纸相关。我们的主要结果表明，对于任何折纸对曲线，存在一个“折纸边缘路径”，这是一系列曲线，$α=α_0，α_1，α_1，α_2，\ cdots，α_n=β$，因此：$α_i$ $α_$α__{i+α_{i+1}任何一对$（α_i，α_j）$都是连贯的；因此，任何填充对，$（α_i，α_j）$也是折纸。建立了它们的存在后，我们提供了最短的折纸边缘路径作为调查领域。
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.
