论文标题
自重曲线曲线和内部边界的组合特性
Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries
论文作者
论文摘要
我们研究了最近定义的最小同拷贝区域概念与自重曲线的经典主题之间的相互作用。后者是平面曲线，是浸入式磁盘边界的图像。我们的第一个贡献是证明新的足够组合条件使曲线是自负的。我们表明，具有惠特尼指数1的曲线$γ$，而没有任何自我重叠的子曲线是自我重叠的。作为推论，我们仅根据曲线及其子曲线的惠特尼指数获得足够的自我布置条件。这些结果取决于我们的第二个贡献，这表明任何平面曲线$γ$（Modulo a basepoint条件）通过用Jordan曲线包裹$γ$将其转化为内部边界。同等地，通过包装，最小同置面积减少到最小的可能阈值，即绕组区域。实际上，我们表明$ n+1 $包装足够，其中$γ$具有$ n $顶点。我们的第三个贡献是证明自我重叠曲线和内部边界的各种定义的等效性，通常是文献中隐含的。我们还介绍并表征了零键曲线，这是由最小同拷贝区域中最优性定义的内部边界的进一步概括。
We study the interplay between the recently defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves which are the image of the boundary of an immersed disk. Our first contribution is to prove new sufficient combinatorial conditions for a curve to be self-overlapping. We show that a curve $γ$ with Whitney index 1 and without any self-overlapping subcurves is self-overlapping. As a corollary, we obtain sufficient conditions for self-overlappingness solely in terms of the Whitney index of the curve and its subcurves. These results follow from our second contribution, which shows that any plane curve $γ$, modulo a basepoint condition, is transformed into an interior boundary by wrapping around $γ$ with Jordan curves. Equivalently, the minimum homotopy area of $γ$ is reduced to the minimal possible threshold, namely the winding area, through wrapping. In fact, we show that $n+1$ wraps suffice, where $γ$ has $n$ vertices. Our third contribution is to prove the equivalence of various definitions of self-overlapping curves and interior boundaries, often implicit in the literature. We also introduce and characterize zero-obstinance curves, further generalizations of interior boundaries defined by optimality in minimum homotopy area.
