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Knots & Classical Electrodynamics

Invited Lecture
by
Samuel J. Lomonaco, Jr. ^(*)
at
Short Course on Knots & Physics
Annual Meeting
American Mathematical Society
San Francisco, California
January, 1995

Abstract

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Samuel J. Lomonaco, Jr.
E-mail: Lomonaco@UMBC.EDU Key words and Phrases. Lord Kelvin, Sir William Thomson, Peter Guthrie Tait, Knots, Electrodynamics, Electromagnetism, Energy, Magnetic Energy, Electrostic Energy, Magnetic Knots, Electrostatic Knots, Minimal Energy Knots, Vortices, Helicity, Magnetic Surfaces, Asymptotic Behavior Objective, Maxwell's Equations, Plasma Physics, Chern-Simon action, Gauge Theory

A Search for a Title

Preamble

Slide 1 The title is ...
Slide 2 On the other hand, the title could have been ...
Slide 3 But the title really should be ...
Slide 4 Acknowledgement

Back to the 19-th Century

Introduction

Slide 0 A letter from J. Clerk Maxwell to Peter Guthrie Tait
Slide 1 The state of electromagnetism before J. Clerk Maxwell
Slide 1a The state of electromagnetism after J. Clerk Maxwell
Slide 2 Conventions adopted during this talk
Slide 3 A chain of events beginning with Herman von Helmholtz
Slide 4 Tait's vortex machine
Slide 4a Tait's lecture with his vortex machine
Slide 5 While attending Tait's lecture, Sir William Thomson (Lord Kelvin) conceived the atomic vortex theory
Slide 6 This fits within the Maxwell Milieu
Slide 7 Another look at Maxwell's letter to Tait
Slide 8 Exactly what was Maxwell saying in his letter?
Slide 9 Tait attempted to correlate knot types with the chemical elements
Slide 9a Tait's Periodic Table
Slide 10 The demise of Thomson's (Lord Kelvin's) atomic vortex theory
Slide 11 Why does the atomic vortex theory still persist?
Slide 12 Why such longevity?

Back to the 20-th Century

Modern Day Magnetic Vortices

Slide 1 Modern day magnetic vortices
Slide 2 Ideal magnetohydrodynamics
Slide 3 Consider an ideal constant density, incompressible, perfectly conducting fluid with a magnetic field B
Slide 4 Equations which govern such a fluid
Slide 5 The magnetic vector potential A
Slide 6 Consequences of incompressibility
Slide 7 The invariance of the volume of a closed surface moving with the fluid
Slide 8 Frozen field effects
Slide 9 Velocity of the magnetic lines of force
Slide 10 Definition of a magnetic surface
Slide 11 Volume is an invariant of a magnetic surface as it moves with the fluid
Slide 12 The magnetic flux enclosed by a magnetics surface is an invariant
Slide 13 The standard torus
Slide 14 The standard foliation of the standard torus
Slide 15 The use of Dehn surgery to construct other foliations of the standard torus
Slide 16 Magnetic knots & links
Slide 17 Two Examples: A magnetic Hopf link and a magnetic trefoil
Slide 18 The boundary of a magnetic knot/link is a magnetic surface. Moreover, it remains a magnetic surface as it changes with the flow.
Slide 19 Some flow invariants: Knot/Link type, enclosed volume, and enclosed flux
Slide 20 Another flow invariant: The linking numbers LK_ij(g_tL)
Slide 21 Another flow invariant: The self-linking number
Slide 22 Summary of flow invariants so far
Slide 23 Another flow invariant: The helicity
Slide 23a The helicity is the same as the Chern-Simon action
- Slide 23b Electromagnetic 1-, 2-, and 3-forms
- Slide 23c Maxwell's equations written in terms of differential forms
- Slide 23d Maxwell's equations in an even more concise form
Slide 24 Helicity as a function of Self-linking and linking numbers
Slide 25 An example: The helicity of a magnetic trefoil from above formula
Slide 26 Helicity of a magnetic Hopf link from above formula
Slide 27 The energy of a magnetic link
Slide 28 Consider a magnetic link in a constant density, incompressible, perfectly conducting, viscous fluid
Slide 29 A description of the energy dissipation of a magnetic knot in a viscous fluid
Slide 30 Energy dissipation terminates because of topology
Slide 31 An example of the energy dissipation of a Magnetic trefoil in a viscous fluid being bounded by topology
Slide 32 How do we quantify the concept of topology bounding energy dissipation?
Slide 32a An informal definition of the asymptotic crossing number
Slide 33 Keith Moffatt's energy spectrum of knots
Slide 34 The energy spectrum of a knot (Cont.)
Slide 35 How do we compute the knot invariants arising from their energy spectrum?

Back to the 20-th Century, Again

Modern Day Electrostatic Vortices

Slide 1 Knotted electrostatic vortices
Slide 2 Conventions assumed during the talk
Slide 3 A charged knotted wire assuming minimal energy position. THONG!
Slide 3a A possible example of a minimal energy electrostatic link
Slide 3b A possible example of a minimal energy electrostatic knot
Slide 4 The honey jar problem
Slide 5 Types of honey jar problems
Slide 6 Types of honey jar problems (Continued)
Slide 7 The Asymptotic Behavior (AB) Objective
Slide 8 Electrostatic energy
Slide 9 The honey jar problem for curves
Slide 10 But the electrostatic energy of a curve is always infinite!
Slide 11 Renormalization and some equations for minimal e-energy knots
Slide 12 Minimal e-energy equations( Continued)
Slide 13 Asymptotic behavior of the equations
Slide 14 The Asymptotic Behavior Objective (AB) yardstick?
Slide 15 Freedman-He-Wang renormalization
Slide 16 A theorem of Freedman & He
Slide 17 Mobius "energy", a non-physical energy. The AB objective yardstick is abandoned.
Slide 17a A reminder of our original motivation
Slide 18 Theorems of Freeman & He on knots/links of minimal Mobius energy
Slide 19 Knots/links of minimal Mobius energy (Continued)
Slide 20 The honey jar problem for hollow & solid tubes. No need for renormalization.
Slide 21 Minimal energy equations for the conducting honey jar problem for hollow tubes
Slide 22 Minimal energy equations (Continued)
Slide 23 Minimal energy equations for honey jar problem for hollow & solid tubes

And on to the Next Millennium ...

The End

Paper based on this talk: The modern legacies of Thomson's atomic vortex theory in classical electrodynamics, in "The Interface of Knots and Physics," edited by L.H. Kauffman, AMS PSAPM, Vol. 51, Providence, RI (1996), pp. 145 - 166.

In postscript form (10.6MB) (22pages)
In gzipped postscript form (1.2MB) (22 pages)
The paper in PDF form (2.0MB) (22 pages)
Web HomePage for Samuel J. Lomonaco, Jr.

(*) Partially supported by the L-O-O-P Fund.

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