Constructing monotone homotopies and sweepouts

Abstract

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than L, then it can also be contracted monotonously through curves of length less than L. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian 2-sphere through curves of length less than L can be replaced with a monotone sweepout through curves of length less than L. Applications of these results are also discussed.

On the complexity of optimal homotopies

“…We begin by considering two curves forming the boundary of a discrete annulus, and study the homotopy between these boundaries of minimal height. Our article leverages on recent results in Riemannian geometry [10,11], and in particular on a companion article co-authored with Gregory Chambers and Regina Rotman [6] where we prove that in the Riemannian setting, such an optimal homotopy can be assumed to be very well behaved. Firstly, it can be assumed to be an isotopy, so that all the intermediate curves remain simple.…”

“…Indeed, as it was not known if the optimal height homotopy was even monotone, the complexity of the problem was completely open. Since the monotonicity result also holds in more geometric settings [6], a recent paper also examined one natural geometric setting, where the goal is to morph across a polygonal domain in Euclidean space with point obstacles; this work presents a lower bound that is linear in the number of obstacles, as well as a 2-approximation for the arbitrary weight obstacles and an exact polynomial time algorithm when all obstacles have unit weight [4]. The same problem also arises as a combinatorics question in lattice theory as a b-northward migration, where the authors leave monotonicity of such migrations as an open question [3].…”

“…This curve is contractible, and we are interested in computing the optimal height of such a contraction. This is one of the settings considered in [6].…”