Unimodal Category and the Monotonicity Conjecture

TL;DR

This paper characterizes the unimodal category for real-valued functions on the real line and circle, provides algorithms for computation, and investigates the monotonicity conjecture, confirming it in some cases and disproving it in others.

Contribution

It offers a complete characterization and algorithms for unimodal category on $\,\mathbb R$ and $S^1$, and analyzes the validity of the monotonicity conjecture across different spaces.

Findings

Unimodal category characterized for functions on $\,\mathbb R$ and $S^1$.

Algorithms developed for computing unimodal category with finitely many critical points.

Monotonicity conjecture proven true on $\,\mathbb R$ and $S^1$, but false on certain graphs and the Euclidean plane.

Abstract

We completely characterize the unimodal category for functions $f:\mathbb R\to[0,\infty)$ using a decomposition theorem obtained by generalizing the sweeping algorithm of Baryshnikov and Ghrist. We also give a characterization of the unimodal category for functions $f:S^1\to[0,\infty)$ and provide an algorithm to compute the unimodal category of such a function in the case of finitely many critical points. We then turn to the monotonicity conjecture of Baryshnikov and Ghrist. We show that this conjecture is true for functions on $\mathbb R$ and $S^1$ using the above characterizations and that it is false on certain graphs and on the Euclidean plane by providing explicit counterexamples. We also show that it holds for functions on the Euclidean plane whose Morse-Smale graph is a tree using a result of Hickok, Villatoro and Wang.