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
reallife 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 theoristsPieranski among themassume 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, howeverno 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 "intertwistedness" of the
knot. To calculate the writhe, you start
with a 2D projection of the knotessentially 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 oneway 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 writhealso called
the 3D writheby 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
DNAwhich is millions of atoms longhas 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
reorientation. 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
harderand 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 dimensionsthe "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 

Writhing
quantum knots

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