Electromagnetic Field Computation Based On Knot Theory

Topology have many applications in the calculation of electromagnetic fields,in this paper, we study the applications of low-dimensional topology-knot theory in the calculation of electromagnetic fields.Knot theory is a subfield of topology in mathematics.In the past,the study of knot is primarily in the field of mathematics.However,in recent years,knots have been gaining interest outside the field of mathematics in various fields of science and engineering,for example,plasma physics,molecular biology.Knot theory has been applicated in the calculation of electromagnetic fields,and many research achievements have been achieved,such as the anslysis of electromagnetic radiation and scattering properties of the torus-knot antenna,the calculation of energy of some special magnetic field structures.In this paper,the electromagnetic properties of a kind of special form of knot-torus knot are discussed,and the main content and innovative results are as follows:First,in order to analyse the antenna radiation and scattering properties,the surface current distribution of antenna under the excitation of incident or impressed field is needed,which requires solution of the electric field or magnetic field integral equation(EFIE or MFIE).In reference[11],a Electric Field Integral Equation(EFIE) suitable for analysis of toroidally knotted wires has been derived based on the Pocklington Integral Equations for an arbitary curved wire. In this paper,a new EFIE is derived based on another famous Electric Field Integral Equation-Hallen Integral Equation for an arbitary curved wire.Second,for the torus knot wire,proved that its self-inductance also can be calculated by the Neumann formula by making use of the concept of knot's Seifert surface,and gives the formula and numerical analysis.