Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries

Abstract

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.