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Round the twist 10 Nov 01

The secrets of the Universe? Tangled up in your shoelaces? Keith Devlin unravels the quantum mystery of knots

SOMETIMES it's hard enough just to find your shoes in the morning. Now you'll have to get used to the fact that tying the laces can entangle you in aspects of quantum theory.

A Polish physicist has recently made the remarkable observation that, just like matter and energy, knots are quantised. The implications are still not well understood, but it has added a new twist to the branch of mathematics known as knot theory, and may have applications elsewhere. It could help unravel the mysteries of how DNA coils itself into tight tangles to fit inside a cell nucleus. And it might even shed light on quantum physics.

This surprising discovery began when Piotr Pieranski of the Poznan University of Technology in Poland wrote a computer program that takes a given knot and, simulating a process in which the rope shrinks, produces what knot theorists call its "ideal" configuration.

Until recently, mathematicians studying knots never worried about the properties of the rope or string used to tie their knots. They were only concerned with the way the knot wrapped around itself and ignored real-life questions such as whether a particular knot can be constructed in practice. Their mathematical knots were constructed out of string that had no thickness, just as the figures of geometry are constructed from idealised lines with no thickness.

In the real world, of course, thickness makes a difference: for example, there are lots of knots you can tie with a thin length of string that you can't using the same length of thicker garden hose. The crucial factor for tying real knots is not the width of the string, but the ratio of the length to the diameter. The smaller this ratio becomes, the harder it gets to construct a given knot. Below a certain threshold in this ratio, you can't construct the knot at all.

There are different ways to tie any particular knot, and for a given length of string some configurations are easier to tie than others. Knot theorists-Pieranski among them-assume that each knot made of real string can be manipulated into an ideal configuration that minimises the length of string used. It is just an assumption, however-no one has ever proved that each knot has a unique ideal configuration.

Once it had produced an ideal configuration, Pieranski's program then went on to calculate what is called the writhe. This is a number that measures the overall degree of "inter-twistedness" of the knot. To calculate the writhe, you start with a 2D projection of the knot-essentially the shadow it would cast if you were to shine a light on it from a particular angle. Then you examine the way the shadow crosses over itself as you follow it round. Imagine the rope is a one-way street. When you cross over another part of the street, does that other part run from left to right or from right to left? If it's going to the right, count it as +1. If it's going left, count it as -1. Once you've done this for all the crossing points, the total of these numbers gives you the 2D writhe of the knot from that particular angle.

But this view from one angle is not enough to tell you what a knot is really like. Even a very simple knot will look different from different angles. So knot theorists calculate the knot's writhe-also called the 3D writhe-by measuring the 2D writhe from all possible angles and then calculating the average of these values. Since there are infinitely many viewing angles, researchers use computer programs to calculate an acceptably accurate approximation.

Pieranski's program calculated the 3D writhes of all ideal, alternating, prime knots with up to nine crossings. A knot is alternating if the crossings alternate between over and under as you go round the knot. It is prime if you can't split it into two simpler knots.

The computer plotted the knots along the horizontal axis and their writhes as points up the vertical axis. At first glance, the plot of the results looked like a random scatter of dots. But then, being a physicist, Pieranski did something that few mathematicians would think to do. He picked up the printout and looked again from a shallow angle to see if he could detect any pattern in the spread of the dots. What he found amazed him. All the 3D writhe values fell along evenly spaced horizontal rows.

Pieranski thought it might have been a fluke so, with the help of his PhD student Sylwester Przybyl, he ran his program again on a much wider range of knots. For about 200 alternating prime knots with up to 10 crossings, the result was the same: each one's 3D writhe fitted into a particular row. It was like looking at the evenly spaced energy levels of electrons in an atom. Knots, it seems, have their own quantum theory.

The discovery amazed everyone in the field. Since it was made by simulating the ways ideal knots are actually tied, Andrzej Stasiak of the University of Lausanne wondered if the same result would come from using a purely theoretical approach. Together with mathematician Corinne Cerf of the Free University of Brussels, Stasiak showed that it could. Cerf and Stasiak proved that the 3D writhe of ideal knots has to be quantised, and that the quantum for knot writhe is 4/7. It fits perfectly with Pieranski and Przybyl's analysis: the 4/7 quantum provides the best fit for the 200 writhes they worked out.

So far, no one really understands why the 3D writhes of ideal knots seem to prefer particular quantised values, nor what is special about the number 4/7. Even so, Pieranski does seem to have stumbled upon a deep and fundamental property of knots that no one had even suspected might be there. His result is to be published in a forthcoming issue of The European Physical Journal E. Cerf and Stasiak have already published their theoretical work in the Proceedings of the National Academy of Sciences (vol 97, p 3795).

Knot researchers are now working to unravel the implications of the mysterious quantum writhe. It could be hugely significant, because knots form an integral part of the processes of life, and may even be at the heart of how our Universe works.

DNA, for example, routinely gets knotted. That's because a copy of an organism's DNA-which is millions of atoms long-has to fit inside the nucleus of each cell. To do that it "supercoils", winding itself up like a telephone cord, and during this process the strands get tangled and knotted. When the DNA is copied it has to be uncoiled, because reading off its genetic information requires untangled and unknotted DNA.

Enzymes called topoisomerases do this job. They free the strands by slicing through crossover points, pulling the two ends free from the knot and then reconnecting the straightened strands. Quantisation of knots makes this complex process much harder. DNA isn't like an ordinary rope, where you can join two ends at any orientation. Its structure, a double helix, means the ends cannot be connected unless they match up in exactly the right way. If the knots in DNA had integer writhe, cutting and rejoining them would be a simple matter, Pieranski says. Integer writhe would mean that the cut ends could be matched together without any re-orientation. But because the writhe numbers are clustered around multiples of 4/7, the enzymes usually have to cut a given DNA knot in several different places in order to undo the supercoiling. In short, the quantum nature of knots means that nature has to work much harder-and more slowly -to manipulate DNA.

Writhe quantisation could also have important consequences for physics, though they may prove much harder to pin down. Some theorists trying to unify Einstein's theories of gravity with quantum mechanics postulate that everything in our Universe is made of tiny vibrating loops coiled up in higher dimensions-the "strings" of string theory. Crucially, these tiny filaments can be knotted, and that means knot theory may have something to say about why the Universe looks the way it does. Quantum theory itself might even be a product of the quantum nature of knots.

At the moment, the idea is wild speculation. What's more certain is that when it comes to understanding knots, the road ahead almost certainly has more twists and turns. The quantum nature of knots was a surprising discovery whose implications are still largely unknown. "It glitters," Pieranski says. "But is it gold?"

Quantum knots
Quantum knots

Writhing quantum knots
Writhing quantum knots

Keith Devlin is a mathematician at Stanford University, California. His latest book is The Maths Gene, published by Phoenix, 7.99

From New Scientist magazine, vol 172 issue 2316, 10/11/2001, page 40