Self-Intersecting Periodic Curves in the Plane
Suppose a smooth planar curve $γ$ is $2π$-periodic in the $x$ direction and the length of one period is $\ell$. It is shown that if $γ$ self-intersects, then it has a segment of length $\ell- 2π$ on which it self-intersects and somewhere its curvature is at least $2π/(\ell - 2π)$. The proof involves the projection $Γ$ of $γ$ onto a cylinder. (The complex relation between $γ$ and $Γ$ was recently observed analytically by T. M. Apostol and M. A. Mnatsakanian. When $γ$ is in general position there is a bijection between self-intersection points of $γ$ modulo the periodicity, and self-intersection points of $Γ$ with winding number 0 around the cylinder. However, our proof depends on the observation that a loop in $Γ$ with winding number 1 leads to a self-intersection point of $γ$.