I would like to solve the following problem (it comes from Morris W. Hirsh, *Differential Topology*, it's exercise 6 section 4 chapter 1):

Show that there is a $C^\infty$ map $f:D^3\to D^2$ with $0\in D^2$ as a regular value such that $f^{-1}(\lbrace0\rbrace)$ is a knotted curve (as in the picture below).

I have been thinking about this problem for a while but I still have no answer. Here is what I came up with so far: if there was no knot on the figure and we wanted $f^{-1}(\lbrace0\rbrace)$ to be a straight line from north to south pole, then $f$ exists, we can take $f$ to be the orthogonal projection onto $(z=0)$. If we do so, we could precompose $f$ by a diffeomorphism of $D^3$ taking the straight line to the knot on the figure, which would give the answer. But I am pretty sure that such a diffeomorphism doesn't exists (the fundamental group of the complement of both path are not isomorphic), and it would be what makes this exercise difficult.

I've been thinking about moving the path to get a better viewpoint, but it didn't succeed.

I am really intrigued by this question, I like it a lot because it doesn't seem right. I am not looking for a complete answer (yet), I just would like a hint to feel how somebody should approach this problem. Thanks in advance for your help.

**Edit:** As suggested by Laz in the comments, there might be an answer to this problem involving techniques such as in this post. The idea would be to construct $f$ with polynomial coefficients.
I went back to read the introduction of the book and this is what M.W.Hirsh says: *"The more challenging exercises are starred, as are those requiring algebraic topology or other advanced topics."* (This is a one star exercise).

So maybe M.W.Hirsh thought about a solution involving polynomial equations etc, but I must admit I would be a bit disappointed in this case, I was hoping that there is a solution involving differential topology. For example I had the following idea: take a tubular neighborhood of the curve $K$, which looks like $I\times D^2$, and define $f$ on this neighborhood by the projection of the second factor. We could try to extend $f$ on $D^3$ (but I don't see how).

Anyway at that point any kind of answer (involving differential topology or not) would be greatly appreciated.

**Edit 2:** Many thanks to HerrWarum for the bounty.