On the groundstate energy of tight knots

TL;DR

This paper analytically determines the minimal magnetic energy of tight knots in an ideal fluid, relating it to ropelength and framing, and compares results with previous studies for various knot types.

Contribution

It introduces a new exact formula for the groundstate energy of magnetic knots based on ropelength and framing, validated with numerical data.

Findings

Analytical expression for minimum magnetic energy as a function of ropelength and framing.

Comparison of torus knot energies with previous work by Chui & Moffatt.

Detailed energy spectrum for prime knots up to 10 crossings.

Abstract

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross-section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm developed by Pieranski (Pieranski, 1998) and collaborators. Results for torus knots are compared with previous work done by Chui & Moffatt (1995), and the groundstate energy spectrum of the first prime knots (up to 10 crossings) is presented and analyzed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.