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.