A General Method for Deriving Vector Potentials Produced by Knotted Solenoids

TL;DR

This paper introduces a general method to derive exact vector potentials for arbitrarily knotted solenoids using magnetostatics, with explicit examples for trefoil and figure-eight knots, relevant for quantum effects like Aharonov-Bohm.

Contribution

It presents a novel, general approach to compute vector potentials for complex knotted solenoids, expanding analytical tools in magnetostatics and quantum physics.

Findings

Explicit expressions for trefoil and figure-eight knots

Method applicable to any knotted solenoid

Relevance to Aharonov-Bohm effect in complex geometries

Abstract

A general method for deriving exact expressions for vector potentials produced by arbitrarily knotted solenoids is presented. It consists of using simple physics ideas from magnetostatics to evaluate the magnetic field in a surrogate problem. The latter is obtained by modelling the knot with wire segments carrying steady currents on a cubical lattice. The expressions for a 31 (trefoil) and a 41 (figure-eight) knot are explicitly worked out. The results are of some importance in the study of the Aharonov-Bohm effect generalised to a situation in which charged particles moving through force-free regions are scattered by fluxes confined to the interior of knotted impenetrable tubes.