The optimal twisted paper cylinder

Abstract An embedded twisted paper cylinder of aspect ratio λ is a smooth isometric embedding of a flat λ × 1 cylinder into ℝ 3 such that the images of the boundary components are linked. We prove that for such an object to exist we must have λ > 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.