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

This paper enumerates and classifies the isotopy classes of degree three Morse polynomials from $\mathbb{R}^3$ to $\mathbb{R}$, proving there are exactly 37 classes with nonsingular principal homogeneous parts and 2,258 classes of strictly Morse polynomials with eight real critical points, utilizing a combinatorial computer program based on Morse surgeries and Picard-Lefschetz theory.

The Gist

Imagine you are a landscape architect, but instead of designing gardens, you are designing the shape of the universe itself. Your tools are mathematical functions—specifically, polynomials that take three-dimensional space (like a room with length, width, and height) and turn it into a single number (like a height map).

This paper by V.A. Vassiliev is a massive cataloging project. The author asks a simple but incredibly deep question: "How many fundamentally different ways can we shape a 3D landscape using a specific type of mathematical rule (a degree 3 polynomial) so that the hills and valleys are perfectly smooth and well-behaved?"

Here is the breakdown of the paper using everyday analogies.

1. The Terrain: Hills, Valleys, and "Morse" Landscapes

Imagine a 3D terrain.

• Peaks are local maximums (hills).
• Pits are local minimums (valleys).
• Saddles are passes between hills (like the dip between two mountain peaks).

In mathematics, a "Morse" landscape is one where these features are perfectly distinct. There are no flat plateaus, no sharp cliffs, and no weird "monkey saddle" shapes where three valleys meet. Every peak, pit, and saddle is a clean, standard shape.

The author is looking at landscapes defined by cubic equations (degree 3). These are complex shapes that can twist and turn in 3D space.

2. The Goal: Counting the "Isotopy Classes"

The paper isn't just asking, "How many hills are there?" It's asking, "How many distinct types of landscapes exist?"

Two landscapes are considered the same "type" (or isotopic) if you can morph one into the other without tearing the fabric of space or creating a new peak or valley out of thin air. It's like playing with clay: you can squish, stretch, and twist the shape, but you can't cut a new hole in it or glue two pieces together.

The author wants to count every single unique "clay sculpture" possible under these rules.

3. The Two Main Families: Type Ξ1 and Type Ξ2

The author discovers that all these cubic landscapes fall into two distinct "families" based on their underlying structure (specifically, how the "horizon" of the landscape looks).

• Family Ξ1: These landscapes have a specific geometric "signature."
• Family Ξ2: These have a different signature.

Think of these like two different species of birds. They might look similar, but their internal anatomy (the math behind them) is different, and they cannot transform into one another.

4. The Big Discovery: The Count

After years of work and using a powerful computer program, the author counts the unique shapes:

• Total Unique Landscapes: There are exactly 37 distinct types of these smooth cubic landscapes.
• The "Perfect" Landscapes: If we demand that the landscape has the maximum possible number of features (8 distinct critical points: peaks, pits, and saddles), there are 2,258 distinct variations.

To put this in perspective: If you tried to list every possible smooth 3D cubic shape, you would need a list of 37 items. If you only care about the ones with the most complex features (8 points), you need a list of over 2,000 items.

5. The Tool: The "Virtual Morse Function"

How did he count them? He didn't build physical models. He invented a digital fingerprint system.

Imagine every landscape has a "DNA." The author created a way to write down this DNA as a grid of numbers (a matrix) and a string of symbols.

• The Matrix: It records how the hills and valleys "talk" to each other mathematically.
• The Surgery: He simulated "surgical operations" on these landscapes. What happens if two hills merge? What if a valley splits? He wrote a computer program to simulate these changes.

By starting with a few known landscapes and letting the computer perform every possible "surgery" (merging, splitting, flipping), he mapped out the entire "family tree" of these shapes. The computer found that the family tree splits into exactly 37 distinct branches.

6. The "Chirality" Twist: Left-Handed vs. Right-Handed

One of the most fascinating findings is about chirality (handedness).

• Some landscapes are achiral: If you look at them in a mirror, they are identical to the original. You can rotate the mirror image to match the original.
• Some landscapes are chiral: They are like your left and right hands. They look similar, but you cannot rotate one to match the other. They are distinct "twins."

The paper identifies exactly which of the 37 shapes are chiral and which are achiral. For example, one specific complex shape (with 8 critical points) has a "mirror twin" that is mathematically distinct, doubling the count for that specific category.

7. The "Passport" of a Landscape

To describe these shapes simply, the author uses a "passport."
A passport is just a list of four numbers: (Minima, Index 1, Index 2, Maxima).

• How many pits?
• How many saddles of type A?
• How many saddles of type B?
• How many peaks?

For example, a passport of (1, 3, 3, 1) means the landscape has 1 pit, 3 saddles of one kind, 3 of another, and 1 peak. This simple list helps categorize the complex shapes.

Summary

Vassiliev's paper is a comprehensive atlas of 3D mathematical terrain.

1. He defined the rules for "smooth" cubic landscapes.
2. He built a computer program to simulate how these landscapes can change shape.
3. He discovered that despite the infinite ways to twist the math, there are only 37 fundamental types of these landscapes.
4. He mapped out the "mirror twins" (chiral pairs) and provided a complete list of every possible variation, especially those with the maximum number of hills and valleys.

It's a bit like discovering that while there are infinite ways to fold a piece of paper, there are only a specific number of ways to fold it into a perfect origami crane without tearing it. This paper lists every single one of those "cranes" for 3D cubic shapes.

Technical Summary

1. Problem Statement

The paper addresses the classification of isotopy classes of Morse polynomials $f: \mathbb{R}^3 \to \mathbb{R}$ of degree three. Specifically, it focuses on polynomials with non-discriminant principal homogeneous parts.

• Context: In bifurcation theory, the classification of connected components of the complement of discriminants (singular zero sets) and caustics (non-Morse critical points) is a central problem. While the classification of smooth zero-level sets (discriminants) for degree 3 surfaces in $\mathbb{R}^3$ is known, the classification of the Morse functions themselves (the "caustic" problem) is significantly more complex.
• Definitions:
  • A polynomial is Morse if all its real critical points are non-degenerate (Morse).
  • It is strictly Morse if, in addition, all critical values at real critical points are distinct.
  • The principal homogeneous part is non-discriminant if its zero set defines a smooth curve in $\mathbb{C}P^2$.
• Types: The space of such polynomials is divided into two connected subsets based on the topology of the cubic curve defined by the principal part in $\mathbb{R}P^2$:
  • Type $\Xi_1$: The principal part defines a cubic curve with one connected component.
  • Type $\Xi_2$: The principal part defines a cubic curve with two connected components.
• Goal: Enumerate all isotopy classes for both types, distinguishing them by the number of real critical points (ranging from 0 to 8, as the sum of Milnor numbers is 8).

2. Methodology

The author employs a sophisticated combination of combinatorial topology, singularity theory, and computer-assisted enumeration.

A. Virtual Morse Functions

The core tool is the concept of a Virtual Morse Function, which encodes the topological data of a generic polynomial:

1. Intersection Matrix: An $8 \times 8$matrix of intersection indices of vanishing cycles in the homology of the complex level set$V_f = f^{-1}(0)$.
2. Real Intersection String: Indices of vanishing cycles with the real locus $V_f \cap \mathbb{R}^3$.
3. Parity String: The parity of the Morse indices of all real critical points.
4. Critical Value Counts: The number of negative and positive critical values.

B. Virtual Surgeries

The classification is achieved by studying elementary virtual surgeries ($s_1$ through $s_7$) that model topological changes in the space of polynomials:

• $s_1, s_2$: Collision of neighboring real critical values (merging or swapping order).
• $s_3, s_4$: Collision of complex conjugate critical values (entering or leaving the real domain).
• $s_5, s_6$: Jumping of real critical values across 0.
• $s_7$: Changes in the system of paths to non-real critical values.

These surgeries define a formal graph where vertices are virtual Morse functions and edges are surgeries. The connected components of this graph are called virtual components.

C. Combinatorial Computer Program

A custom computer program (based on previous work in [33], [36]) was used to:

• Generate all possible virtual Morse functions starting from a base polynomial.
• Perform chains of virtual surgeries to explore the formal graph.
• Count the number of elements in each virtual component (the Card invariant).
• Reconstruct integer Morse indices from parity data using specific intersection criteria (Proposition 14).

D. D-Graphs

For polynomials with all real critical points, the author introduces the D-graph (a directed, colored graph derived from the Coxeter-Dynkin diagram). The D-graph serves as a complete invariant for isotopy classes with 8 real critical points, equivalent to the set-valued virtual component.

3. Key Contributions and Results

A. Main Enumeration Theorems

The paper provides a complete enumeration of isotopy classes:

1. Total Isotopy Classes (Morse Polynomials):

   • Type $\Xi_1$: 21 isotopy classes.
   • Type $\Xi_2$: 16 isotopy classes.
   • Total: 37 isotopy classes.
   • Breakdown: The classes are distributed by the number of real critical points (0, 2, 4, 6, 8) as detailed in Table 1 of the paper.

2. Strictly Morse Polynomials (8 Real Critical Points):

   • Type $\Xi_1$: 842 isotopy classes.
   • Type $\Xi_2$: 1416 isotopy classes.
   • Total: 2258 isotopy classes.
   • Topological Property: Each of these 2258 isotopy classes is homotopy equivalent to the group $SL(3, \mathbb{R})$.

B. Singularity Splitting (Theorems 4 & 5)

The paper investigates which pairs of singularities can appear as critical points of a single degree-3 polynomial (where the sum of Milnor numbers is 8).

• Existence: Proves the existence of specific pairs for $\Xi_1$ (e.g., $E_6+A_2$, $D_5+A_3$, $A_4+A_4$) and $\Xi_2$ (e.g., $A_5+A_3$, $D_5+A_3$).
• Non-existence: Proves the impossibility of other combinations (e.g., $A_7+A_1$, $D_4^+ + D_4^-$) using Euler characteristic arguments and D-graph splitting constraints.

C. Chirality and Symmetry

• Achirality: Most isotopy classes are achiral (invariant under orientation-reversing affine transformations).
• Chirality: One specific virtual component for Type $\Xi_1$ (Card=1233, D-graph in Fig. 9) is chiral, meaning it splits into two distinct isotopy classes that are mirror images of each other.

D. Realization

The author explicitly constructs polynomial examples (using normal forms like Hessian and Newton-Weierstrass) that realize every identified D-graph and virtual component, proving the completeness of the classification.

4. Significance

• Extension of Local Theory: This work extends the classification of local perturbations of simple singularities (Arnold's ADE classification) to the global classification of degree-3 polynomials in $\mathbb{R}^3$. It solves the "parabolic" case ($P_8$ singularities) in three variables.
• Complexity of Higher Dimensions: It demonstrates that the classification of Morse polynomials is significantly more complex than the classification of their zero sets (nonsingular surfaces). While nonsingular cubic surfaces have a known classification, the space of Morse functions has 37 distinct topological types.
• Methodological Advancement: The paper refines the "virtual Morse function" technique, adapting it from 2D (where integer Morse indices can be directly predicted) to 3D (where only parities are initially known, requiring reconstruction algorithms).
• Topological Insight: The result that the space of strictly Morse polynomials with 8 critical points is homotopy equivalent to $SL(3, \mathbb{R})$ provides deep insight into the topology of the configuration space of these functions.

5. Conclusion

Vassiliev successfully enumerates all 37 isotopy classes of degree-3 Morse polynomials in $\mathbb{R}^3$ with non-discriminant principal parts. By utilizing a combinatorial computer program to analyze virtual surgeries and D-graphs, the paper resolves the classification of both general Morse and strictly Morse cases, revealing a rich structure of 2258 strictly Morse classes and establishing the homotopy type of their parameter spaces.