by Jonathan Kujawa
One hundred and fifty years ago atoms were mysterious things. They could only be studied indirectly. We knew about their interactions with each other as a gas, the frequencies of light they prefer to absorb and emit, and various other properties. Nowadays we can capture the image of a single hydrogen atom, but back then atoms could only be understood through the shadows they cast in the macro world.
At the time two explanations were in vogue. The atomists went with the ancient Greeks and viewed atoms as small billiard balls clacking against each other as they moved through empty space. This point of view worked great for explaining the behavior of gases, but didn't help much in explaining the intrinsic properties observed by chemists. On the other hand, the followers of the theory of Boscovich, an eighteenth century Jesuit, thought that atoms were points of force which alternately repelled and attracted each other depending on how close they were. This theory held promise for explaining the electromagnetic properties of atoms, but it also had its drawbacks.
On February 18, 1867 William Thomson (aka Lord Kelvin) read out his paper “Vortex Atoms” to the assembled members of the Royal Society of Edinburgh. In it he suggested a novel alternative to these two theories.
As everyone knew at the time, the universe was permeated by the luminiferous ether. Light traveled as a wave even through “empty” space and, well, waves travel through something, so what was that something? Luminiferous ether! It was a beautiful idea, but eventually the evidence piled up against the ether. The Michaelson-Morley experiment put a stake through its heart in 1887.
But in 1867 the luminiferous ether was widely considered a standard feature of the physical world. Taking his inspiration from recent work in hydrodynamics and, presumably, a fine pipe of tobacco, Lord Kelvin realized that instead of viewing atoms and the ether as two separate things, we could instead think of atoms as vortices in the ether itself. Specifically, he thought of each atom as a knotted tubular shape:
His theory neatly explained a wide variety of atomic phenomena. The rich variety of possible knots justified the wide variety of atoms, the fact that the type of knot is unchanged under small perturbations (after all, you can't turn the knots in Lord Kelvin's table from one into another without applying real violence) explains the robust stability of atoms, and knots will clearly vibrate at different frequencies from one another and so will naturally prefer to absorb and emit light energy at differing levels. For example, Thomson thought the two linked circles in the lower left might be the sodium atom because of sodium's two spectral lines.
All in all, Vertex Atoms looked to be a real winner. Thomson's case found support in the work of his friend and collaborator, Peter Tait. In January of 1867 Tait set up an apparatus which allowed him to study the behavior of various smoke rings.
Thomson was one of those in attendance as they played with (sorry, researched 🙂 these vortices. Peter Jackson reenacted Tait and Thomson's investigations in a recent film.
The beauty and effectiveness of atomic vortices made Thomson's theory compelling. Unfortunately, the confirmation of the nonexistence of the luminiferous ether was also the death knell for atomic vortices.
Nevertheless, Tait found knots to be irresistibly interesting. Tait began to compile a complete list of all the different possible knots, where here and elsewhere we mean knots in the sense of Lord Kelvin. Namely, Tait was interested in completely describing all possible knotted closed loops where we only care about their “knottiness”. If one knot can be obtained from another by moving strands and, crucially, never making any cuts, we will consider them to be the same knot.
As is often the case, a question this easy to ask is devilishly hard to answer. But this also makes the question full of potential. Lord Kelvin himself appreciated the richness of the subject:
A full mathematical investigation of the mutual action between two vortex rings of any given magnitudes and velocities passing one another in any two lines, so directed that they never come nearer one another than a large multiple of the diameter of either, is a perfect mathematical problem; and the novelty of the circumstances contemplated presents difficulties of an exciting character.
“Difficulties of an exciting character,” indeed! Through much labor Tait first classified all knots which can be drawn with seven crossings (the minimal number of crossings required when drawing a particular knot is called the crossing number of the knot). Later, Tait built on the work of others and provided a complete classification of all knots of ten or fewer crossings. To get a taste for how daunting a task this was, you can take a glance at all the knots with ten or fewer crossings .
Or consider the following knot designed by Kauffman and Lambropoulou. They call it The Culprit:
Can strands be moved to and fro to simplify it to a knot we already know? Amazingly enough The Culprit can be simplified down to a single closed loop (the “unknot”). It is a fun/maddening puzzle to transform this guy into the unknot. It might be helpful to make The Culprit out of an actual piece of string . A hint: you actually have to increase the number of crossings while simplifying.
In an attempt to better understand and, ultimately, classify all knots, mathematicians have developed a number of invariants of knots. After all, it would be nice to be able to identify The Culprit as the unknot without the torture of having to actually figure out how to untangle it. A knot invariant is something computable which gives you the same answer regardless of how the knot is drawn.
There is an inherent tension here: if an invariant could distinguish between all possible knots, then it would necessarily be just as complicated as the knots themselves and, presumably, no easier to deal with. The key is to strike a balance between ease of use and effectiveness. In the past century we've come up a number of knot invariants: the crossing number, tricolorability, the writhe, and the Alexander-Conway polynomial, to name a few. See Carl Pierer's essays for 3QD on the closely related topic of braids (here and here).
With the demise of the luminiferous ether and Lord Kelvin's theory of atomic vertices, knots lost some of their appeal. Despite being a rich source of tantalizing questions the study of knots was something a boutique subject, even within mathematics. There was an interest in developing knot invariants for curiosity's sake, but the subject wasn't seen as particularly close to the rest of mathematics (never mind the rest of science).
In 1984 lightning struck. Vaughn Jones announced the discovery of a new knot invariant based on polynomials. It is now called the Jones polynomial and its discovery revolutionized mathematics. Indeed, he won the Fields medal in 1990 for his discovery.
The Jones polynomial was the first major new knot invariant discovered in years. Much more importantly, it illuminated deep connections between knot theory, other areas of mathematics, and physics. Quantum groups, Khovanov homology, Floer homology, Chern-Simons theory, and topological quantum field theory have all played a role in this grand new understanding. And what could be argued to be the first topic of twenty-first century mathematics, categorification, is a direct descendent of Jones's polynomial.
Now this is all well and good, but why should non-mathematican, non-sailor types care about knots? In what has to be some sort of record for scientific Second Comings, Lord Kelvin's atomic vortices have made something of a comeback. The fundamental idea behind modern string theory is that subatomic particles are in fact tiny little strings. Just as with atomic vortices, the properties of the particles are determined by how the strings are knotted and how they vibrate. It's no surprise, then, that the well known physicist and string theorist, Edward Witten, has a keen interest in knot theory.
We also now know that understanding how a protein folds is a fundamental question in biology. Misfolded proteins are believed to be a key cause of Alzheimer's disease, Parkinson's disease, cystic fibrosis, and many other degenerative disorders. Knot theory and knot invariants play a useful role in studying which sorts of foldings are possible.
There is a problem, though. Mathematicians have, by and large, studied knots from Tait and Thomson's original point of view that their “knottiness” is all that matters. These are idealized, Platonic sorts of knots where the strands can be stretched, twisted, squeezed as much as you like. Anything short of cutting is a-okay in the knot theory game.
But knots in real life are made of shoelaces, proteins, and rope. A rope has thickness and length and this can't be ignored. It's going to be awfully hard to tie The Culprit if all you have is a few inches of thick rope. As you can imagine, adding the constraint of length and thickness only makes the problem of knots that much harder. But there are intrepid souls who are willing to tackle even these questions. This area is called geometric knot theory. Geometric because you are interested in studying knots while taking into consideration issues of geometry like length and angle.
For example, you can ask how “bendy” your knot is by measuring the distortion. Distortion measures how far you have to travel to go from point to point staying on the knot versus if you allow yourself to leave the knot to take shortcuts. A very bendy knot will have lots of places where the journey along the knot is much longer than a shortcut through space. In the mid-eighties Gromov asked if there was some absolute bound to how distorted a knot could be. So far nobody knows!
I first learned about geometric knot theory from Jason Cantarella. Jason is a mathematician at the University of Georgia with a keen interest in the sorts of problems which arise when you mix the flexibility of knots and curves with the rigidity of geometry. In addition to geometric knot theory, Jason has written papers on the square peg problem we came across here at 3QD and on linkages. Linkages are knots made with straight line segments with hinged “elbows”. They show up in robotics, the folding of airbags in automobiles, the unfolding of solar panels in NASA spacecraft, etc.
Another geometric question you can ask is for the ropelength of your knot: with a rope of a given thickness , what is the shortest length you can use to tie that knot? For the linked circles Lord Kelvin thought might be sodium it turns out that that that ropelength is eight times pi.
The knot in the upper left corner of Lord Kelvin's illustration is called the trefoil knot. It is second only to the unknot for simplicity. Amazingly, we don't know its ropelength. Our best estimates put it between 31.32 and 32.743175. So far nobody knows the true value!
There are various techniques which often seem to give close to the shortest possible ropelength, but it is darn difficult to know that you couldn't possibly do better. If you are a practical person of little curiosity, you could settle for using a 33 foot rope to tie your trefoil knot. But on your deathbed you'll wonder if you could have done better.
Jason has used computer searches to find knots which are near to minimal in their ropelength. His software, Ridgerunner, is freely available on his website if you'd like to give it a go. If nothing else, it's quite mesmerizing to watch the videos made by Jason of Ridgerunner's best attempt at a minimal ropelength for each knot.
In March here at 3QD we came across the work of Henry Segerman. Henry likes to use 3D printing to make the sorts of mathematical models Tait and Thomson could only dream of. Henry has a nice video where he talks about knots and, in particular, how he used Jason's calculations to 3D print minimal ropelength versions of all the knots with seven or fewer crossings.
It's remarkable how much and how little we know 150 years after Lord Kelvin first wondered if there was something more to smoke rings.
 Image borrowed from a nice article in Quanta magazine which explains how we've come full circle and are now using knot theory to understand vertices in fluid flow.
 Image borrowed from here where they built a modern version of Tait's vortex cannon.
 To be precise, we are only classifing the “prime” knots. Some knots can be obtained by combining two simpler knots by cutting the two simple knots and gluing the strands together to merge them into one bigger knot. Those that can't be made in this way are, by analogy with the prime numbers, called prime knots. Once you know these, the rest can be made by gluing.
 Real knot theorists never wear loafers.
 We'll assume our ropes are always of radius one unit.
 Thanks to “Topology and Physics – a Historical Essay” by Charles Nash and “William Thomson: Smoke Rings and Nineteenth-Century Atomism” by Robert Silliman for some of the history surrounding Lord Kelvin and his Vortex Atoms.