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By Harold Henderson
UIC mathematics professor Louis Kauffman took off his belt and handed one end to me. I held it still, the fabric straight and flat between us. He turned his end over--one, two, three, four times clockwise. The belt twisted along its length. "That's 720 degrees," he said--two complete 360-degree turns. Then Kauffman passed his end just once counterclockwise underneath the rest of the belt, changing hands in the process but keeping the end pointed toward himself. I thought, surely he's going to double it up into an awful tangle. Instead the twists all vanished. The belt lay as flat and straight as when we began.
The belt trick is no optical illusion. Like airplane flight or Michael Flanagan's being elected to Congress, it's an incredible but undeniable fact--a paradox. No matter how often or how well explained, it still seems weird.
But it's all in a day's work for Kauffman, one of the world's leading practitioners of knot theory, a subset of topology, the mathematical study of space. Knot theorists don't limit themselves to knots. Kauffman takes a professional interest in anything that twists, tangles, links, or knots in two or more dimensions.
Those of us who are neither mathematicians nor poets tend to prefer our paradoxes out of sight and out of mind. But Kauffman seeks them out. Then he scrutinizes them, plays with them, tries to find parallels in other paradoxes. Some of the underlying patterns he discovers will become useful in biology, chemistry, physics. Some will simply remain oddities. Of course you can't do much math if you worry about which are which, and you can't tell ahead of time anyway.
* * *
The belt trick certainly looks like an oddity. In fact it seems like magic, and that feeling is not new. For millennia people have felt superstitious awe at knots, twists, and tangles. Many cultures believed that the right knot at the right time could bind the wind, inhibit fertility, and cause disease or cure it. In 1718 a resident of Bordeaux was sentenced to be burned alive for bewitching a local family with knotted cords.
Kauffman--who uses the belt trick as a mathematical teaching tool, a sort of performance mathematics--says the belt trick is not an isolated paradox. Rather it follows a distinct pattern--a pattern that happens to be repeated in the Philippine wine dance.
You can almost do this dance sitting down. Professionals perform it while holding a glass in the palm of one hand, but to avoid breakage you can leave the glass on the table. Just hold your right arm up with the palm flat and hand pointed back like a waiter hoisting a tray, thumb next to your head, pinkie finger on the outside. Rotate your hand counterclockwise in a half circle and bring it down to waist level so that it's pointing forward. Then turn it halfway around again. So far that's a 360-degree turn--your hand should be pointing backward again with thumb close and pinkie outside--but it's not the position you started in. Now keep turning your hand counterclockwise while moving it out from your body and up and forward. Your palm remains flat (if your joints are more flexible than mine) and your hand points forward again at forehead level, a bit awkwardly, elbow akimbo. Finally rotate the hand back toward and over your head to its original position--720 degrees from the beginning. You've done it, and, as Kauffman says, "Your arm isn't broken." Performance math strikes again.
The pattern in both the belt trick and the wine dance can be partially described mathematically using the number i, mathematicians' name for the square root of -1. (Multiply i times itself and you get -1.) "You can think of each half twist [or 180-degree turn] as multiplying by i," says Kauffman. The belt starts out flat, and he arbitrarily chooses to call that position 1. "Turn it over once. That's 1 times i, which is the same as just i. Turn it over again. That's i times i, which is -1." That's two twists, or 360 degrees, but you're not back to where you started. "Turn it over a third time; that's -1 times i. Turn it over the fourth time [720 degrees]; that's -1 times i times i, which is -1 times -1, which is 1--where you started."
This multiplication doesn't "explain" the belt trick or the wine dance. But by making the underlying pattern clearer, it makes them seem a little less paradoxical.
For Kauffman, pattern is the point, whether he's looking at the belt trick or a knot. "Mathematics is the study and classification of pattern," he writes. Math lives in a "seemingly airy realm of the study of patterns formed by patterns."
* * *
Halfway up the second tallest building on the campus of the University of Illinois at Chicago, Kauffman's office is jammed with books, journals, file cabinets, puzzles, models of knots and polyhedrons, and a computer terminal. There's room for two other people, as long as they don't make any sudden moves. The one clear surface is a nine-foot-long blackboard along the east wall. In that space the world of physical things meets and mingles with the hidden world of patterns, provided that Kauffman can find the chalk.
He's won UIC's University Scholar award for teaching and research, and he's held visiting professorships at universities in Swansea, Kyoto, Bologna, Berkeley, and Zaragoza. Along with W.B.R. Lickorish of Cambridge and M. Wadati of the University of Tokyo, he edits the four-year-old Journal of Knot Theory and Its Ramifications. He's also working on a series of technical books published by Singapore-based World Scientific Publishing, called the "Series on Knots and Everything." (In what passes for a joke among mathematicians, a colleague told Kauffman that if the books were about everything, then adding "knots" to the title was redundant.) He wrote volume one, the 723-page Knots and Physics; coedited volume three, Quantum Topology; and edited volume six, Knots and Applications.
Kauffman first took to math when he was in junior high, living near the Canadian border in Norfolk, New York, and reading Martin Gardner's Scientific American column Mathematical Games. In the late 1950s he built an electrical gadget that played ticktacktoe as a science project. It's in his office at UIC too--a brown rectangular board with white lettering and dials on the front, a black coil of wires behind. He had to connect the switches himself rather than rely on modules or printed circuits. "I think it was more fun when you had to understand exactly what you were doing."
And exactly what is Kauffman doing now? Studying patterns, yes. But he's studying particular patterns with a particular attitude. By analogy, using numbers to count beans may involve patterns, but it's really just accounting. Mathematics begins when you start wondering about the numbers themselves. A bean counter might notice that 17 can be divided evenly only by itself and 1. But he probably wouldn't make lists of numbers like 17, or give them a name--prime numbers--or wonder how many there are. Mathematicians do all these things, even though the ultimate objective may be obscure.
In the same way, knot theory begins whenever someone goes beyond tying shoes and boats and horses, and instead starts wondering about the knots themselves--both the obvious patterns and the less obvious underlying ones. How many are there? How can you tell them apart? What makes them "knotted" anyway? Can you take them apart without untying them? What's the shortest length of rope in which you can tie a knot? Can you describe them with a mathematical formula? Can you turn each one into its mirror image?
* * *
Knots are beyond ancient. Knot theory is younger than Chicago. It didn't grow out of mathematical soil first, but out of a seeming paradox that puzzled Sir William Thomson, Baron Kelvin, one of the great minds of 19th-century physics. Physicists back then explained the durability of matter by assuming that it was made up of identical, indestructible, marblelike atoms. But in that case, wondered Kelvin, how do groups of atoms hold together in molecules? And how can gases be compressed and expand?
In 1867 he proposed an alternative, a "kinetic theory of gases." (Kauffman reprinted the original article in Knots and Applications.) According to this theory, the atoms making up gases are not like marbles; they're more like persistent knotted smoke rings in the ether (an all-pervading substance then postulated to exist by scientists). Kelvin expected that his colleagues would develop similar theories for solids and liquids. Then each element would be made up of "vortex atoms" that were knotted in a unique way to produce the different properties of hydrogen and iron, chlorine and chromium.
If Kelvin's theory were true, then a table of the elements would also be a table of unique knots. No such table existed in 1867, but suddenly it seemed important to develop one. Imagine the embarrassment if you identified oxygen as knot A and manganese as knot B, and later discovered that A and B were the same knot!
Kelvin's knotted atoms were soon superseded, though the end of this story hasn't been written--Kauffman says that the current quantum-theory description of atoms bears "a strong resemblance" to Kelvin's notion. But the idea of building and studying a table of knots had taken hold in pure mathematics.
* * *
The belt trick is a blatant paradox. The paradoxes of knots are subtler, perhaps because their patterns are far more intricate. For instance, theorists have long known that there's only one knot with three crossings, but not until December 1994 did the Journal of Knot Theory publish the news that there are 19,536 different 14-crossing knots that "alternate" (always go over and then under). The "Jones polynomial" excited knot theorists when it was discovered in 1984, because it allowed them to distinguish between many knots and their mirror images--but there are some ten-crossing knots it can't tell apart, and it's not yet clear whether it can tell a knot from a mere tangle.
Even the simplest knots have an elusive quality, says Kauffman. No matter how hard you stare at them or how often you tie them, it always seems like you might be missing something. After all, if you'd never seen it done, would you believe that two limp pieces of string could adhere to each other just by being wrapped together in a certain way?
Kauffman and his colleagues approach knots much the way they do the belt trick--by looking for patterns. But because knots are so complicated, they use several different methods. Kauffman likes to switch back and forth from blackboard diagrams to a length of rope in his hands to his Macintosh laptop. "I have all these different languages I think about them in."
But the length of standard gray rope in his pocket is his first language. On more than one occasion when I expected an esoteric blackboard equation, he brought out the rope and showed me how the knot worked in 3-D. The simplest of all knots, the "trefoil," has three crossings.
Yet even the simple trefoil is easier to classify and study if its two-dimensional "shadow" can be frozen in a standard format on paper. Imagine tying a trefoil in a length of rope, and then bringing the two ends of the rope together and fusing them into a closed loop. The knot is now preserved like a butterfly pinned to a board (of course you can't tie it or untie it, but as a knot theorist you don't care). Now project the shadow of this loop on paper and trace it, leaving little gaps to show which strand crosses on top at each crossing. Voila--a diagram, the knot theorists' favorite way of looking at knots. (Among other things, it enables them to define the word: a "knot" is any closed loop that you can't pull, twist, stretch, or bend into a plain circle without cutting the rope.)
Now you can make like the accountant who became a mathematician: forget for a minute that this diagram represents a three-dimensional piece of rope. What about the two-dimensional diagram itself? You can see it as simply three curved line segments.
Of course the knot isn't really three disconnected lines. But thinking about it that way can lead to discovering new patterns in it. "That's a lot of what we do in topology," says Kauffman. "We study spaces by breaking them up into their building blocks." Even apparently seamless, continuous things like the trefoil knot.
Knot theorists also study how one space can be embedded in another. If this sounds like double-talk, Kauffman asks you to imagine a rope tied in a trefoil knot and frozen into an ice cube with its ends sticking out. The knot is already a space embedded in another space, but that fact may be easier to see if you imagine one more step: dissolve the rope chemically without disturbing the ice. The knot shape remains in the ice cube--a space embedded in another space. "Usually when we look at a knot," he says, "we put ambient space in the background. As a topologist I try to see the knot and the space it's in at the same time."
Knot theorists also label knots mathematically, identifying each with a numerical (or algebraic) code called an invariant. Over the years knot theorists have devised many invariants, each named for its discoverer: Alexander, Conway, Jones, Homfly, Kauffman. Many invariants are computed by looking closely at each crossing in the knot, calculating a score for it, and then putting the scores together. All should remain the same no matter how a knot is pulled, bent, or twisted--hence the name. A good invariant will quickly tell you whether a knot is really knotted, for instance, or whether it can be deformed into its mirror image.
* * *
Sometimes all this analysis and labeling has a real-world payoff. Sometimes it doesn't. Mathematicians don't necessarily care what their search for patterns and paradoxes might accomplish in the rest of the world; that's part of what makes them mathematicians. When I first stepped into Kauffman's office, I imagined knot theory as a narrow academic side road--intriguing on its own terms, like checkers, but not leading much of anywhere.
That might have been true in the 1930s. But these days knot theory has become the intellectual equivalent of Interstate 80, with interchanges that lead almost anywhere. Knots now intersect with organic chemistry, the biology of DNA, quantum physics--maybe even metaphysics. The problems these disciplines bring to Kauffman's door are "not necessarily the sort of problems a topologist would think of," he reflects. "But the mathematics involved is not trivial. And thinking about the applications leads to new mathematical problems as well"--just as Kelvin's vortex atoms led to the "pure" mathematics of knot theory, which has now begun to reconnect with physics.
Synthetic chemistry and knot theory are also feeding off each other, as David Walba and colleagues at the University of Colorado, Boulder, describe in one chapter of Knots and Applications. Synthetic chemists are in the business of creating new molecules to form new kinds of chemicals, so they're always on the lookout for models to follow. First they cribbed from topology, making molecules modeled on basic knots and on the Mobius strip. If you missed out on this topological paradox as a kid, cut a piece of paper into a ribbon, give it one half twist and tape the ends together. The resulting closed loop has only one side and one edge. Walba's group was the first to create a molecule shaped like this. Its creation alerted topologists to new mathematical facts about it--and now the chemists are trying to build molecules based on complex knots.
Another case: The two strands of the DNA packed into our cells can be twisted around each other or linked together like two rings. These phenomena interest knot theorists, but what really gets them going is that in order to replicate, these strands have to break and then recombine with different ends than before. Biologists can't watch the enzymes that break the strands, but they can see (just barely) how the DNA is arranged before and after. From these snapshots topologists try to deduce what the enzymes must have been doing. In a draft paper titled "Gap Migration and a Possible Role of Knot Dynamics in the Process of RNA Splicing," Kauffman and Yuri Magarshak (of the consulting and research firm Biology & Technology International) show that certain kinds of RNA--single-stranded molecules transcribed from DNA--may tie themselves into particular knots in the process of changing from one form to another--a job that would no doubt be terminally baffling if they hadn't already counted and classified all knots up through 14 crossings. (Even in this technical paper the authors encourage their academic readers to switch "languages" when it helps: "The transformations described above look rather tricky in writing. But if one performs them with elastic cable, they become very understandable.")
As far as we know, there are no knots in basic physics. But there do seem to be mathematical parallels. In the early 1980s, by a process Kauffman describes as "a mix of utter stupidity and flashes of intelligence," he discovered a new knot invariant that describes knots using equations very much like the ones physicists use in statistical mechanics to track elementary particles. The pattern that's seen in the belt trick also reappears in basic physics, in a more complex version of the "i
Taking these convergences another step, Kauffman even speculates that the math of quantum mechanics may be directly applicable to the math for DNA--in which case, "knot theory, quantum field theory, and molecular biology are fundamentally one scientific subject."
And then he goes even further and connects knot theory to metaphysics, using the paradox of the Mobius strip. "Its properties go against common sense and yet they are true," he writes in a recent paper. Even though it looks like an ordinary loop of ribbon, it's not--a fact you can verify by drawing a path down the middle of the ribbon. Your pencil will cover both "sides" and come back to the beginning without ever crossing an edge. "Locally, an observer on the band will see two edges and detect two sides. If the observer walks along the band he/she will eventually return to her starting place, but will find that she is on the other side of the band!" She must go around again to get back to the starting point.
Once again Kauffman embraces the paradox rather than backing away, as if to answer the age-old question, is the world made up of many things, as it appears to be, or is it just one? "The band embodies a form of which it may be said that it is ONE (one side) and yet it is also MANY (two sides). . . . To speak this way in the absence of the model would seem to be [self-contradictory], yet in the Mobius band we have a picture before us of just how this unity in multiplicity can come about. It is ONE for the global observer and MANY for the local observer. It is ONE for the local observer who is willing to travel and experience the whole. It is MANY for the global observer who is willing to play the game that looks locally and forgets for a moment the whole." It's not every philosopher who can slap together a model of the nature of the universe from scrap paper at a moment's notice.
Art accompanying story in printed newspaper (not available in this archive): photographs by Randy Tunnell.