The Optimal Twisted Paper Cylinder

An embedded twisted paper cylinder of aspect ratio $\lambda$ is a smooth isometric embedding of a flat $\lambda \times 1$ cylinder into ${\mathbb{R}}^3$ such that the images of the boundary components are linked. We prove that for such an object to exist we must have $\lambda >2$ and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to $2$ must converge to a (non-smooth) $4$-fold wrapping of a right-angled isosceles triangle.