Transparencies Used in Lecture
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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
© Copyright 1995
COPYRIGHT STATEMENT
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Samuel J. Lomonaco, Jr.
Email:
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, ChernSimon 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 19th 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 20th 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_{t}L)
 Slide 21
Another flow invariant: The selflinking 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 ChernSimon action
A digression: Maxwell's equations expressed in terms of
differential forms
 Slide 23b
Electromagnetic 1, 2, and 3forms
 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 Selflinking 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 20th 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
eenergy knots
 Slide 12
Minimal eenergy equations( Continued)
 Slide 13
Asymptotic behavior of the equations
 Slide 14
The Asymptotic Behavior Objective (AB) yardstick?
 Slide 15
FreedmanHeWang renormalization
 Slide 16
A theorem of Freedman & He
 Slide 17
Mobius "energy", a nonphysical 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.
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(*) Partially supported by the LOOP Fund.