Measure of a 2-component link

TL;DR

This paper introduces a geometric measure for two-component links based on symplectic integration, identifying minimal configurations as the optimal Hopf links, thus connecting link theory with symplectic geometry.

Contribution

It establishes a novel geometric measure for links using symplectic forms and characterizes minimal links as Hopf links, linking topology with symplectic geometry.

Findings

The measure equals the torus area in the configuration space.

Minimal measure is achieved only by Hopf links.

The configuration space relates to cotangent bundles and Grassmannians.

Abstract

A two-component link produces a torus as the product of the component knots in a two-point configuration space of a three-sphere. This space can be identified with a cotangent bundle and also with an indefinite Grassmannian. We show that the integration of the absolute value of the canonical symplectic form is equal to the area of the torus with respect to the pseudo-Riemannian structure, and that it attains the minimum only at the "best" Hopf links.