自重叠曲线与内边界的组合性质
Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries
摘要：
我们研究了最近定义的最小同伦面积的概念和自重叠曲线的经典主题之间的相互作用。后者是平面曲线，是浸没圆盘边界的图像。我们的第一个贡献是证明新的充分组合条件的曲线是自重叠的。我们证明了Whitney指数为1且没有自重叠子曲线的曲线$γ$是自重叠的。作为推论，我们得到了仅用曲线及其子曲线的Whitney指数表示的自相似性的充分条件。这些结果遵循我们的第二个贡献，这表明，任何平面曲线$γ$，模基点条件，被转换成一个内部边界的包裹$γ$与约旦曲线。等价地，通过缠绕将$γ$的最小同伦面积减小到最小可能阈值，即缠绕面积。事实上，我们表明，$n+1$包装足够，其中$γ$有$n$顶点。我们的第三个贡献是证明自重叠曲线和内部边界的各种定义的等价性，往往隐含在文献中。我们还介绍和零固执曲线的特点，进一步推广的最优性定义的最小同伦区域内的内部边界。
英文摘要：
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.
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