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

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.