A closed regular plane curve $\gamma$ with positive non-constant curvature $\kappa(s)>0$ and rotation index (turning number) $1$. Does it implies $\gamma$ is simple? Note that rotation index is computed as total curvature divided by $2\pi$.

This seems to me intuitive clear. But I cannot prove it. Can anyone help me to handle this?

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    $\begingroup$ If the curve intersects itself, it then is naturally divided into two closed curves (likely with corners). So what will the rotation index be? $\endgroup$ Jun 14, 2021 at 3:03


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