A Proof of the Invariant Based Formula for the Linking Number and its Asymptotic Behaviour

TL;DR

This paper provides a detailed proof of an explicit invariant formula for the linking number of curves, originally introduced in 2000, and explores its asymptotic behavior, enhancing understanding of its geometric properties.

Contribution

It offers the first rigorous proof of the linking number formula based on six isometry invariants and analyzes its asymptotic behavior, which was previously unstudied.

Findings

Proof of the linking number formula based on six invariants

Analysis of the formula's asymptotic behavior

Enhanced understanding of geometric invariants in 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.