Regarding a converse to Hopf's Umlaufsatz

Question

I read in a differential geometry textbook that the total signed curvature of a closed plane curve is an integer multiple of $2\pi$. In that same textbook, I also read about Hopf's Umlaufsatz, which states that the total signed curvature of a simple closed plane curve is either $2\pi$ or $-2\pi$. Now, I am interested in a converse of the previous statement. If a closed plane curve has total signed curvature either $2\pi$ or $-2\pi$, must that curve be a simple closed plane curve?

Answer 1

The "total signed curvature", as you call it, is an isotopy invariant for smooth, oriented, closed plane curves (i.e. immersions of $S^1$ in $\mathbb{R}^2$). The answer to your question is "no" because you can deform a simple closed curve into a nonsimple closed curve. So, for instance, this curve is isotopic to a simple counter-clockwise loop and has the same total signed curvature.

Here's a slightly more interesting-looking example, this time isotopic to a simple clockwise loop.

In general, the total signed curvature is a complete isotopy invariant. If two immersions of $S^1$ have the same total signed curvature, then they are isotopic. That is, one can be deformed to the other through a family of immersions.