Ribbon invariants I

TL;DR

The paper introduces a new invariant called the ribbon invariant $b3$ for smooth functions on a 2-disk, which estimates the number of geometrically distinct critical points and offers a combinatorial alternative to traditional degree-based methods.

Contribution

It defines and analyzes the ribbon invariant $b3$, providing estimates, algorithms for computation, and exploring its connections to geometry and graph theory.

Findings

$b3$ estimates the number of essential critical components.

Algorithms for calculating $b3$ are developed.

Connections to immersed curves and graph theory are discussed.

Abstract

A non-negative integer invariant, estimating from below the number of geometrically different critical points of a smooth function $f$ defined in the 2-disk, $f:\mathbb{B}^{2}\rightarrow\mathbb{R}$, is considered. (We denote it by "$\gamma$".) It depends on combined $C^{0}+C^{1}$ type conditions on the boundary $\partial(\mathbb{B}^{2})=\mathbb{S}^{1}$, that we call "ribbons" here. It turns out to be an alternative to the degree of the gradient map and almost independent from it. Note that the computation of the degree does not guarantee multiple critical points, unlike the ribbon invariant $\gamma$. In fact, this invariant is counting the number of essential components of the critical set, rather than simply the number of critical points. Various estimates of $\gamma$ are established. Some other "ribbon type" invariants of geometrical nature are defined and investigated. All these invariants turn out to be more combinatorial, rather than algebraic, in nature. Algorithms for the calculation of the ribbon invariants are presented. Interconnections with some different areas, such as the theory of immersed curves in the plane or independent domination in graphs, as well as various geometric applications, are commented. The latter topics will be investigated in detail in Part II of the present article. At the end, different questions about $\gamma$ are asked.