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The Physics of Knots

From Smoke Rings to Cosmic Strings

[Talk slides here]

I decided to make this page with a few links to things I reference in my knots talk. As I ad lib a lot of the talk I can't guarantee any of these things were mentioned - in particular, Cosmic Strings rarely get a mention despite being in the title!

The key players are: Peter Guthrie Tait, who did the first serious work in golf ball aerodynamics (see, for example, here) and smoke rings blown out of Space Raiders crisp boxes; William Thomson, later Lord Kelvin, Tait's friend whose circulation theorem explains the stability of the smoke rings; and James Clerk Maxwell, friend to both, whose theory of light as electromagnetic waves helped inspire Kelvin's theory of atoms as knots in the luminiferous aether (which we now know to be incorrect). An excellent introduction to the story of these three friends' development of Knot Theory can be found in Daniel Silver's Essay Scottish Physics and Knot Theory's Odd Origins.

In showing how difficult it can be to distinguish two knots I gave the example of the Perko Pair, two knots believed to be different but proven in 1974 to be the same. In particular the two knots shown are often incorrect, and actually are different! A history of this story is available here.

The Borromean Rings are three rings where all three are linked but any two aren't (if the third is removed). You might like to think about making a four ring version, or even an n-ring version for n>3. They remind me of the Incompatible Food Triad, a problem whose history is outlined by George Hart here. The problem is to find three foods where any two don't go together but all three do (or more commonly the counter-problem, to find any three foods which do go together but all three don't). I've heard some good suggestions so far. My favourite, given by my friend Matt N. K. Smith, is Water, Whiskey, and Ice. It also reminds me of Roger Penrose's impossible tribar, which he himself relates to what are known as mathematical `cohomology cycles' in this paper. The Borromean Rings are certainly similar to Efimov States in atomic physics, and far less convincingly similar to Sartre's No Exit.

Knots are used to symbolise infinity in the Eternal Knot in Buddhism (note the topologically distinct knots used!), the Ourobouros, and arguably the usual infinity symbol, especially in the form used by Gauss which looks a lot like a circular ribbon (unknot) with a half-twist in it. The name for the symbol is a lemniscate, meaning `decorated with ribbons'.

Examples of knots in physics which I sometimes state are: the vortices coming off the wings of planes; Wilson Loops in quantum field theory; knotted DNA strands especially in Bacteria; and braided particles in topological quantum computers. The example I usually fail to remember to mention is Cosmic Strings. By themselves these aren't usually so knotty, but if you find two moving in the correct way and fly around them in the correct way you get a time machine - effectively this is the `Thumb Trick' (which I explain in the Physics World article on this page), but instead of your thumb rotating the hand on your clock rotates. Backwards. I provide some slightly more detailed notes from a talk I've given to scientific audiences here. Oh, and getting out of armlocks.