A formula for the linking number in terms of isometry invariants of straight line segments

The linking number is usually defined as an isotopy invariant of two non-intersecting closed curves in 3-dimensional space. However, the original definition in 1833 by Gauss in the form of a double integral makes sense for any open disjoint curves considered up to rigid motion. Hence the linking number can be studied as an isometry invariant of rigid structures consisting of straight line segments. For the first time this paper gives a complete proof for an explicit analytic formula for the linking number of two line segments in terms of six isometry invariants, namely the distance and angle between the segments and four coordinates of their endpoints in a natural coordinate system associated with the segments. Motivated by interpenetration of crystalline networks, we discuss potential extensions to infinite periodic structures and review recent advances in isometry classifications of periodic point sets.