find a closed curve with given winding numbers

Question

Is there a closed (piecewise) $C^1$ curve $\gamma: [a,b]\rightarrow\mathbb{C}$, so that $\mathbb{C}\setminus\gamma([a,b])$ has four components with winding numbers -1, 0, 2, 3?

Thanks!

Answer 1

Here is a solution (see graphics below), $\gamma([a,b])$ is the concatenation of three (or better said four) circles:

The circuit begins in $B$, describes twice the circle with center $C$, then describes the circle with center $D$, all of them with a positive orientation, then, arriving back in $B$, describes the circle with center $A$ in the negative direction.

Here is a parameterization with $a=0, b=4$: $$\gamma(t)=\begin{cases} -2+2e^{2i \pi t}& (0 \leq t<2)\\ -1+e^{2i \pi t}& (2 \leq t <3)\\ 2-2e^{-2i \pi t}& (3 \leq t <4) \end{cases}$$

(thanks to @Daniel Fischer who has detected a bug in these intervals)