April 2001

# Knotty problems

Anyone who has tried to save their Christmas tree lights to use the next year will know that, no matter how carefully you put them away, as soon as you start to uncoil them, you find that at best you are faced with some nasty looking knots and at worst a hopeless tangle.

Neither science nor mathematics can explain why this always happens, and probably never will. But, thanks to some recent work by the mathematicians Jeffrey Lagarias of AT&T Laboratories and Joel Hass of the University of California at Davis, we do now know how much effort might be involved in straightening out the lights.

The new result is in knot theory, a branch of mathematics that goes back to the work of Gauss in the middle of the 19th century. It began with a typical pure mathematician's question: Can you find a mathematical way to describe different knots?

One way to try to classify knots is by counting the number of times the string (or Christmas tree lights) crosses itself, and much of the early work in knot theory consisted of discovering how many genuinely different kinds of knot there are for each given "crossing number."

It turns out that there is one knot with crossing number 3, one with crossing number 4, two with 5, three with 6 crossings, seven with 7, twenty-one with 8, forty-nine with 9, and one hundred and sixty-five with 10. In 1998, using computers, two teams of knot theorists managed to tabulate all knots having 16 or fewer crossings. There are exactly 1,701,936 of them altogether.

One reason why knot theory is such a tricky subject is that it's hard to see if two knots are the same simply by looking at them. For instance, stage magicians often present you with what looks like a knotted rope, but then they pull on the two ends and, lo and behold, the "knot" simply falls away. The rope wasn't really knotted at all; it was just tangled up.

In fact, this is precisely the issue that Lagarias and Hass's new result addresses. Given a length of string that is all tangled up, but not technically knotted, what is the maximum number of steps it could take you to straighten it out?

Before you try this, I should tell you that, as with the magician's "knot", you are not allowed to let go of the free ends. In fact, mathematicians eliminate the free-ends issue altogether by assuming that the string is a closed loop, something that could be achieved in practice by gluing the two ends together.

With the free ends gone, the only way to untie a knot, or untangle a tangle, is by manipulating the string. But how? Knots theorists have long known that a combination of just three basic moves -- called Reidemeister moves after their discoverer -- will always untangle a tangled up loop that is not actually knotted. The problem is, no one knew just how many Reidemeister moves it could take.

Now, thanks to the recent theorem we do. But don't hold your breath. According to Lagarias and Hass's result, if the loop has N crossovers, then you can untangle it in no more than 2 to the power (100 billion times N) basic moves. That would take way longer than the entire life of the universe. So the result is not practical. On the other hand, going from infinite to any finite number is a big breakthrough, and it could set the stage for finding a more realistic number.

So who cares? Does this result have any practical importance? Other than the importance of knots to scouts, campers, and sailors, does knot theory have any uses anyway? As so often happens in mathematics, what started as a question driven by pure curiosity has turned out to be of major importance in at least two sciences.

Physicists now believe that matter is made up of tiny loops of space-time -- the strings of Superstring Theory --- and the mathematics of knot theory turns out to be exactly what they need to describe those loops.

Secondly, knot theory plays a role in our understanding of DNA. Because a typical DNA molecule is so long, it has to coil itself up to squeeze into the cell. Some viruses work by changing the knot structure of the DNA, making it behave differently (to benefit the virus rather than the original "owner" of the DNA). By using electron microscopes and the mathematics of knots, collaborative teams of biologists and mathematicians have started to make headway in figuring out just how some viruses manage to infect and take over a cell - knowledge that might one day lead to more effective medicines to fight disease.

With applications like those, almost any advance in our understanding of knots could turn out to have enormous significance. Meanwhile, knot theorists are delighted to have found a bound on the number of moves it takes to straighten out a tangled unknot. Even if it won't help them at Christmas.

Devlin's Angle is updated at the beginning of each month.

A slightly different version of this article appeared in The Guardian newspaper in the UK in March.

Keith Devlin (devlin@stmarys-ca.edu) is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books