Energies of knot diagrams

TL;DR

This paper introduces new knot energies on diagrams that model physical properties of metallic tori, analyzing their mathematical properties and normal forms using advanced techniques like elliptic functions.

Contribution

It defines a new family of knot energies based on physical models and analyzes their minimizers using the Gauss representation and elliptic functions.

Findings

Identification of normal forms as minima of the energy functional

Development of a mathematical framework for knot energies related to physical models

Application of elliptic functions to characterize energy minimizers

Abstract

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function $f$ and study the simplest nontrivial case $f(x)=x^2$, obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.