1. A Proof of the Invariant Based Formula for the Linking Number and its Asymptotic Behaviour
- Author
-
Bright, Matt, Anosova, Olga, and Kurlin, Vitaliy
- Subjects
Mathematics - Geometric Topology - Abstract
In 1833 Gauss defined the linking number of two disjoint curves in 3-space. For open curves this double integral over the parameterised curves is real-valued and invariant modulo rigid motions or isometries that preserve distances between points, and has been recently used in the elucidation of molecular structures. In 1976 Banchoff geometrically interpreted the linking number between two line segments. An explicit analytic formula based on this interpretation was given in 2000 without proof in terms of 6 isometry invariants: the distance and angle between the segments and 4 coordinates specifying their relative positions. We give a detailed proof of this formula and describe its asymptotic behaviour that wasn't previously studied., Comment: Accepted for publication in Springer Lecture Notes in Computational Science and Engineering with a presentation at NUMGRID 2020 Conference, http://www.ccas.ru/gridgen/numgrid2020. Version 3 is extended for publication in conference proceeding - all technical proofs are now in the main body of the paper
- Published
- 2020