Geometric considerations for energy minimization of topological links and chainmail networks

TL;DR

This paper explores the geometric configurations that minimize energy functions in topological links, revealing insights into optimal structures like Hopf links, Borromean rings, and chainmail networks, with implications for knot theory and material design.

Contribution

It introduces a method to find minimal energy embeddings of complex links and chainmail structures, and proposes a new universal lower bound for knot and link ropelength.

Findings

Minimal energy configurations for Hopf links and Borromean rings identified.

Japanese-style chainmail networks are more energy-efficient than lattice-based ones.

A conjectural improved lower bound for knot and link ropelength proposed.

Abstract

Knot and link energies can be computed from sets of closed curves in three dimensional space, and each type of knot or link has a minimum energy associated with it. Here, we consider embeddings of links that locally or globally minimize the M\"obius and Minimum Distance energies. By describing these energies as functions of a small number of free parameters, we can find configurations that minimize the energies with respect to these parameters. It has previous been demonstrated that such minimizers exist, but the specific embeddings have not necessarily been found. We find the geometries leading to minimal configurations of Hopf links and Borromean rings, as well as more complex structures such as chain links and chainmails. We find that scale-invariant properties of these energies can lead to ``non-physical'' minimizers, e.g. that a linear chain of Hopf links will subtend a finite length as its crossing number diverges. This incidentally allows us to derive a conjectural improved universal lower bound for the ropelength of knots and links. We also show that Japanese-style square chainmail networks are more efficient, in terms of excess energy, than square lattice ``4-in-1'' chainmail networks.