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 $\gamma$ 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 $\gamma$, modulo a basepoint condition, is transformed into an interior boundary by wrapping around $\gamma$ with Jordan curves. Equivalently, the minimum homotopy area of $\gamma$ is reduced to the minimal possible threshold, namely the winding area, through wrapping. In fact, we show that $n+1$ wraps suffice, where $\gamma$ 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.

中文翻译：

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自重叠曲线与内边界的组合性质

我们研究了最近定义的最小同伦面积概念与自重叠曲线的经典主题之间的相互作用。后者是平面曲线，它是浸入盘边界的图像。我们的第一个贡献是证明曲线自重叠的新的充分组合条件。我们表明，惠特尼指数为 1 且没有任何自重叠子曲线的曲线 $\gamma$ 是自重叠的。作为推论，我们仅根据曲线及其子曲线的惠特尼指数就获得了自重叠的充分条件。这些结果来自我们的第二个贡献，它表明任何平面曲线 $\gamma$，以基点条件为模，通过用 Jordan 曲线环绕 $\gamma$ 转换为内部边界。同样，通过包裹将$\gamma$的最小同伦面积减小到最小可能的阈值，即缠绕面积。事实上，我们证明 $n+1$ 就足够了，其中 $\gamma$ 有 $n$ 个顶点。我们的第三个贡献是证明了自重叠曲线和内部边界的各种定义的等效性，这些定义通常隐含在文献中。我们还介绍和表征了零顽固曲线，进一步概括了由最小同伦区域中的最优性定义的内部边界。