A principle for ideal torus knots

TL;DR

This paper introduces a new principle for determining ideal torus knots as maximally rotated, zero-twist structures, enabling rapid calculations and comparison with biological molecules.

Contribution

It proposes a novel method to identify ideal torus knots through maximally rotated configurations, bypassing traditional rope shortening routines.

Findings

Ideal structures can be found as maximally rotated, zero-twist configurations.

The method allows rapid numerical computation of ideal torus knots.

Comparison with biological torus molecules shows relevant aspect ratios.

Abstract

We study simple, knotted and linked torus windings that are made of tubes of finite thickness. Knots which have the shortest rope length are often denoted ideal structures. Conventionally, the ideal structure are found by rope shortening routines. It is shown that alternatively they can be directly determined as maximally rotated structures. In many cases these structures are also zero-twist structures i.e. structures that neither rotate one or the other way under strain. We use this principle to implement rapid numerical calculations of the ideal structures and subsequently quantify them by their aspect ratio. The results are compared with the aspect ratios of biological torus molecules.