Let $C$ be a curve in the plane whose curvature is everywhere $\le 1$. If $C$ has length $L$, what is the largest number of proper self-crossings of $C$ as a function of $L$?
For example, the curve below has length $L=2 \pi (n + \epsilon)$ and has $n(n-1)$ proper crossings, where $n=5$ in the figure. So this pattern achieves $L (L-2\pi) /(4 \pi^2))$ crossings as $\epsilon \to 0$, and so grows quadratically in $L$.
Q1. Can anyone see a pattern that improves on the above curve? E.g., can more than $20$ intersections be achieved with $L \approx 10\pi$?
Q2. Can the number of crossings grow faster than $L^2$?