TITLE: Height function on 2-torus with only 3 critical points
QUESTION [39 upvotes]: It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number by Lusternik-Schnirelmann theory): a minimum, a maximum, and a degenerate saddle.
It is not hard to describe these functions by means of their levelsets, but it seems difficult to produce immersions of $T^2$ into $\mathbb R^3$ with a height function that does the job. According to Banchoff and Takens, there are no smooth embeddings with such height functions, only immersions. I was trying to look for pictures of such immersions and came across the following beautiful image by Cassidy Curtis:

I expected that these would be somehow easier to produce -- but probably I am wrong... still:

Is there a "simpler" immersion of $T^2$ into $\mathbb R^3$ whose image has a height function with exactly 3 critical points? Or is the above example "optimal" in some sense?

The only necessary conditions I see are that the saddle point $p$ must be degenerate and that there are 3 arcs with both endpoints at $p$ that lie in the same levelset as $p$. I find it hard to believe that these conditions are not sufficient, and that there are no easier immersions; but I have not been able to prove this or find any such immersions.

REPLY [10 votes]: Warning: this is not an immersion (it has twelve Whitney-umbrella-like pinch points)
Here is a relatively simple explicit realization: the $z$ coordinate for the parametric surface$$(x,y,z)=(\sin(2u),\sin(2v),\sin(u)\sin(v)\sin(u-v));$$after an affine shift leaving $z$ unchanged the graph looks like this:

It is thus similar to the Steiner's Roman surface except that the latter has three double lines and this one has six.
I've tried to cut it to make the central monkey saddle more visible:

The implicit equation, in slightly different coordinates, is$$(X+Y+Z)(X+Y-Z)(X-Y+Z)(-X+Y+Z)=(XYZ)^2,$$
 
with function $X+Y+Z$:

Mathematica codes:
ParametricPlot3D[
 {3Sin[2u]+4Sin[u]Sin[v]Sin[u-v],3Sin[2v]-4Sin[u]Sin[v]Sin[u-v],Sin[u]Sin[v]Sin[u-v]},
 {u,0,\[Pi]},{v,0,\[Pi]},
 BoxRatios->{1,1,1},
 Mesh->None,
 PlotPoints->150,
 PlotStyle->FaceForm[Red,Cyan],
 Boxed->False,Axes->False,
 SphericalRegion->True
]

ParametricPlot3D[
 {3Sin[2u]+4Sin[u]Sin[v]Sin[u-v],3Sin[2v]-4Sin[u]Sin[v]Sin[u-v],Sin[u]Sin[v]Sin[u-v]},
 {u, 0, \[Pi]}, {v, 0, \[Pi]},
 BoxRatios -> {1, 1, 1}, Mesh -> None,
 RegionFunction -> (Abs[#3]<.05 \[Or] .2<Abs[#3]<.4 \[Or] .6<Abs[#3]<.8 \[Or] Abs[#3]>1 &),
 PlotPoints -> 250,
 PlotStyle -> FaceForm[Red, Cyan],
 BoundaryStyle -> Black,
 Boxed -> False,
 Axes -> False,
 SphericalRegion -> True
]

With[{d=2},
 ContourPlot3D[
  (U-V-W) (U+V-W) (U-V+W) (U+V+W)+(U V W)^2==0,
  {U,-d,d},{V,-d,d},{W,-d,d},
  BoxRatios->{1,1,1},
  PlotPoints->150,
  MeshFunctions->{#1+#2+#3&},
  ImageSize->Full,
  Mesh->40
 ]
]