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.

Topological Picturebook. Looks almost like Francis drew them. $\endgroup$ – Ryan Budney Feb 2 '16 at 8:33