Athwart curves in \({\mathbb{R}}^ 3\) (Q1073351)

Two closed curves in \({\mathbb{R}}^ 3\) are called athwart if no tangent line to one curve intersects any tangent line to the other. The authors prove three theorems for smooth (seemingly meaning \(C^ 3)\) athwart curves. The linking number of any two athwart curves as well as the linking number of one curve relative to a tangent line of the other are zero. The Grassmann images (Gauss image projected into \({\mathbb{P}}^ 2)\) of two athwart curves coincide. The inverse image of any point of the Grassmann image of an athwart couple cannot be a single point. As a consequence, any closed space curve without a couple of parallel tangent lines (whose existence was proved by B. Segre) cannot be athwart to any curve.