A spectral approach to the linking number in the 3-torus

TL;DR

This paper introduces a spectral method to compute the linking number of multi-curves in a 3-torus, connecting topological linking with the spectral theory of the Laplace operator on differential forms.

Contribution

It provides a new spectral formula for linking numbers in the 3-torus, extending previous results to more general multi-geodesics.

Findings

Derived a spectral formula relating linking number to Laplace operator eigenvalues

Computed linking numbers for multi-geodesics in the 3-torus

Generalized Dehornoy's result to higher dimensions

Abstract

Given a closed Riemannian manifold and a pair of multi-curves in it, we give a formula relating the linking number of the later to the spectral theory of the Laplace operator acting on differential one forms. As an application, we compute the linking number of any two multi-geodesics of the flat torus of dimension 3, generalising a result of P. Dehornoy.