A closed plane curve $\gamma$ with positive non-constant curvature $\kappa(s)>0$ and rotation index $1$. Does it implies $\gamma$ is simple?

Question

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?