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Knots & Classical Electrodynamics
Samuel J. Lomonaco, Jr. (*)
Short Course on Knots & Physics
American Mathematical Society
San Francisco, California
© Copyright 1995
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Samuel J. Lomonaco, Jr.
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
- 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
Back to the 19-th Century
A letter from J. Clerk Maxwell to Peter Guthrie Tait
The state of electromagnetism before J. Clerk Maxwell
The state of electromagnetism after J. Clerk Maxwell
Conventions adopted during this talk
A chain of events beginning with Herman von Helmholtz
Tait's vortex machine
Tait's lecture with his vortex machine
While attending Tait's lecture, Sir William
Thomson (Lord Kelvin) conceived
the atomic vortex theory
This fits within the Maxwell Milieu
Another look at Maxwell's letter to Tait
Exactly what was Maxwell saying in his letter?
Tait attempted to correlate knot types with the chemical elements
Tait's Periodic Table
The demise of Thomson's (Lord Kelvin's) atomic vortex theory
Why does the atomic vortex theory still persist?
Why such longevity?
Back to the 20-th Century
Modern day magnetic vortices
Consider an ideal constant density, incompressible, perfectly
conducting fluid with a magnetic field B
Equations which govern such a fluid
The magnetic vector potential A
Consequences of incompressibility
The invariance of the volume of a closed surface moving with the
Frozen field effects
Velocity of the magnetic lines of force
Definition of a magnetic surface
Volume is an invariant of a magnetic surface as it moves with
The magnetic flux enclosed by a magnetics surface is an
The standard torus
The standard foliation of the standard torus
The use of Dehn surgery to construct other foliations of
the standard torus
Magnetic knots & links
Two Examples: A magnetic Hopf link and a magnetic trefoil
The boundary of a magnetic knot/link is a magnetic surface.
Moreover, it remains a magnetic surface as it changes with the flow.
Some flow invariants: Knot/Link type, enclosed volume,
and enclosed flux
Another flow invariant: The linking numbers
Another flow invariant: The self-linking number
Summary of flow invariants so far
Another flow invariant: The helicity
The helicity is the same as the Chern-Simon action
Modern Day Magnetic Vortices
Electromagnetic 1-, 2-, and 3-forms
Maxwell's equations written in terms of differential
Maxwell's equations in an even more concise form
A digression: Maxwell's equations expressed in terms of
Helicity as a function of Self-linking and linking numbers
An example: The helicity of a magnetic trefoil
from above formula
Helicity of a magnetic Hopf link
from above formula
The energy of a magnetic link
Consider a magnetic link in a constant density,
incompressible, perfectly conducting, viscous fluid
A description of the energy dissipation of a magnetic
knot in a viscous fluid
Energy dissipation terminates because of topology
An example of the energy dissipation of a Magnetic trefoil
in a viscous fluid being bounded by topology
How do we quantify the concept of topology bounding energy
An informal definition of the asymptotic crossing number
Keith Moffatt's energy spectrum of knots
The energy spectrum of a knot (Cont.)
How do we compute the knot invariants arising from their
Back to the 20-th Century, Again
Knotted electrostatic vortices
Conventions assumed during the talk
A charged knotted wire assuming minimal energy position. THONG!
A possible example of a minimal energy electrostatic link
A possible example of a minimal energy electrostatic knot
The honey jar problem
Types of honey jar problems
Types of honey jar problems (Continued)
The Asymptotic Behavior (AB) Objective
The honey jar problem for curves
But the electrostatic energy of a curve is always infinite!
Renormalization and some equations for minimal
Minimal e-energy equations( Continued)
Asymptotic behavior of the equations
The Asymptotic Behavior Objective (AB) yardstick?
A theorem of Freedman & He
Mobius "energy", a non-physical energy. The
AB objective yardstick is abandoned.
A reminder of our original motivation
Theorems of Freeman & He on knots/links of minimal Mobius energy
Knots/links of minimal Mobius energy (Continued)
The honey jar problem for hollow & solid tubes.
No need for renormalization.
Minimal energy equations for the conducting honey jar problem
for hollow tubes
Minimal energy equations (Continued)
Minimal energy equations for honey jar problem for hollow &
Modern Day Electrostatic Vortices
And on to the Next Millennium ...
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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for Samuel J. Lomonaco, Jr.
(*) Partially supported by the L-O-O-P Fund.