How many passes does a monkey saddle have?

Question

John Stillwell writes in Poincaré's Papers on Topology:

The study of "pits, peaks, and passes" on surfaces in $\mathbb R^3$ by Cayley (1859) and Maxwell (1870). A family of parallel planes in $\mathbb R^3$ intersects a surface $S$ in curves we may view as curves of "constant height" (contour lines) on $S$. If the planes are taken to be in general position, and the surface is smooth, then $S$ has only finitely many "pits, peaks, and passes" relative to the height function. It turns out that $$\text{number of peaks} - \text{number of passes} + \text{number of pits}$$ is precisely the Euler characteristic of $S$.

So a peak is a local maximum, a pit is a local minimum. But what is a pass?

From the wiki on mountain pass I guess that a pass is a path from one hollow to another hollow. Say, we have a saddle, then we have one pass.

But what if we have a monkey saddle and thus three hollows adjacent to each other:

How many passes do we have here and what are they exactly?

Edit: From "Surface Topology" by Firby and Gardiner:

A pass moves from a hollow, through the critical point, into the adjacent hollow. Hence, the number of passes through the critical point is one less than the number of hollows adjacent to the critical point.

I guess we have three hollows adjacent to the critical point. So we should have two passes. But what and where are they?

Answer 1

If our surface is given locally by an equation $z=f(x,y)$, then I would think that a "peak" is a critical point where the Hessian has two negative eigenvalues, a "pit" is a point where it has two positive eigenvalues, and a "pass" is a point where it has one negative and one positive eigenvalue.

As such, the critical point of the Monkey saddle, given by $z=x^3-3xy^2$, is none of the above, as its Hessian vanishes at $(0,0)$. Another way of saying it: to a best quadratic approximation, this surface is flat at the origin.

Answer 2

The monkey saddle is not in general position, but there is an interesting thing to do here, and that is to perturb the function. You can check that if we perturb the function $f(x,y)=x^3-3x y^2$ slighly, for example to $g(x,y)=x^3-3x y^2+\epsilon x$ for $\epsilon>0$ small, we will not find one pass, but two (this does not depend on the generic perturbation chosen). One can prove that the Monkey saddle counts as two passes in the count above. This can be made rigorous using Conley theory, or Gromol-Meyer theory. You can also consult my PhD thesis (this is discussed on page 13 of the introduction).