The Point-Split Method and the Linking Number of Space Curves

Abstract: This is a report on research done in the field of mathematical physics. It is an investigation of the concept of the linking number between two simple and closed spatial curves. The linking number is a topological invariant with scientific applications ranging from DNA biology to Topological Quantum Field Theory. Our aim is to study C ̆alug ̆areanu’s theorem, also called White’s formula, which relates the linking number to the concepts of twist and writhe. We are interested in the limit of the two curves as they approach each other. To regulate this, we introduce a regularization method that utilizes a point-split. Further we explore if the result is dependent on how the regularization is introduced. Therefor we inflict an asymmetry in the regularization, with a parameter a in the point-split intervals, to check whether the result becomes dependent on a or not. We find that the result is independent of the parameter a.