Isotopy classification of Morse polynomials of degree 3 in ${\mathbb R}^3$

TL;DR

This paper classifies all isotopy classes of degree three Morse polynomials from R^3 to R^1, identifying 37 classes with nonsingular principal parts and 2258 with maximal critical points, using computational topology tools.

Contribution

It provides a complete enumeration of isotopy classes of degree three Morse polynomials in three variables, incorporating computational methods for Morse surgeries and monodromy.

Findings

37 isotopy classes with nonsingular principal parts

2258 isotopy classes with maximal critical points

Use of computer programs for classification

Abstract

We enumerate all isotopy classes of degree three Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ with nonsingular principal homogeneous parts, proving that there are exactly 37 of them. We also count all 2258 isotopy classes of {\em strictly} Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ of degree three with the maximal possible number (eight) of real critical points. A main tool in this classification is a combinatorial computer program that formalizes Morse surgeries, local monodromy and Picard-Lefschetz theory.