On the Height of a Homotopy

Given 2 homotopic curves in a topological space, there are several ways to measure similarity between the curves, including Hausdorff distance and Fréchet distance. In this paper, we examine a different measure of similarity which considers the family of curves represented in the homotopy between the curves, and measures the longest such curve, known as the height of the homotopy. In other words, if we have two homotopic curves on a surface and view a homotopy as a way to morph one curve into the other, we wish to find the longest intermediate curve along the morphing. We prove that given a pair of disjoint embedded homotopic curves, among minimal height homotopies on the surface, there exists an ambient isotopy; in other words, the homotopy with minimum height never makes a “backwards” move and results in disjoint simple intermediate curves.