To Knot or Not?

How to generate and identify random knots in a piece of string.

Dorian M. Raymer

Editor's Note: Being published in the Proceedings of the National Academy of Sciences (PNAS) is a major achievement, especially when you are an undergraduate student. Recently The New York Times Magazine named student Dorian Raymer's PNAS study on random knot formation, which he wrote with Douglas Smith, an assistant professor of physics at the University of California at San Diego, "one of the Top Science Stories of 2007." The Citizen Scientist asked Dorian if he would be willing to provide us with a summary of his paper, he quickly agreed and we are very pleased to publish it here.

What governs the annoying formation of random knots in strings, cords, and garden hoses? Mathematicians have studied knots for more than a century, but only in the abstract. As physicists, we have also found this to be an interesting physics problem.

What is a knot? How many different knots exist? How do knots form?

The word knot has several meanings. Colloquially, a knot refers to a string that is tangled or interlaced with itself or another strand of rope.

Take a necklace that has a clasp/connector. Unclasp the necklace so that there are two free ends. Tie a simple knot; the first knot you usually tie in your shoe laces will do. Finally, re-clasp/close the necklace. Mathematicians define "knot" as a closed curve that cannot be untangled to produce a simple loop (see www.Mathworld.wolfram.com). A necklace is an example of a closed loop. The knot you tied before you re-clasped the necklace is what most people commonly refer to as a trefoil or “granny” knot. And the re-clasped necklace is what a mathematician would call a knot.

Figure 1. Some examples of knots.

People tie knots all the time. They are useful for keeping shoe laces tight. Doctors use knots to keep stitches in place. And sailors rely on knots to hold masts, booms and sails in place. People have figured out hundreds of ways to tie knots to suit specific purposes; many of them are complicated and difficult to tie.

Knots can also occur out of nowhere, usually undesirably so. For instance, when a person stores long, flexible cord, such as Christmas lights, headphones, or yarn, in a confined area like a bag or a box without exercising extreme care, there is a high probability that cord will be knotted upon retrieval at some later date. These annoying knots are often impressively complex and impossible to untie.

In an experiment recently published in the Proceedings of the National Academy of Sciences (Dorian M. Raymer and Douglas E. Smith, "Spontaneous knotting of an agitated string," PNAS 104, 16432-16437, 16 October 2007), we sought to gain scientific perspective on the phenomenon of "spontaneously formed" knots. We were looking to identify fundamental factors that contribute to causing what some call "Murphy's Law" -- of knotting.

Our experiments consisted of tumbling pieces of string in a rotating box much as clothes are tumbled in a dryer. We repeated this experiment several thousands of times. The string was specially selected and characterized, having a certain length, width and flexibility, and the box rotation was precise and repeatable. The main thing we started investigating was how the probability of a knot
forming varied with the length of string, keeping constant all other parameters, like box size and rotation rate.

The results of initial experiments confirmed an obvious hypothesis that, within some confined volume, longer string (relative to the box size) is more likely to become knotted than shorter string.

Something that was not immediately obvious, but became more apparent as the experiment carried on, was that a fairly large number of different knots of considerable complexity can form by themselves.

How to Identify an Unknown Knot

Over the years, knot theorists have come up with a multitude of methods for identifying and classifying knots. The most straight forward and generally useful one is called the minimum crossing number.

Sticking to the mathematician's definition of a knot leads to the property --one all knots hold-- that a knot is unchangeable. Test this out by using a necklace or a piece of string to make the first non-trivial knot in Robert G. Scharein's knot zoo here: the 3_1, also known as the Trefoil. An important quality about the knots in the knot zoo is that they are prime. Just as the numbers 2,3,5,7. . . cannot be divided into components (like the number 10 can be broken into 2 and 5), prime knots cannot be broken into a combination of other simpler knots.

The knots in the knot zoo are the simplest you can get, and the Trefoil is the simplest knot of all, other than the trivial unknot that has zero crossings. Tinkering with the string/necklace should give you a stronger sense for why this is so.

You should also see that, although the minimum crossing number is 3, it is possible to arrange the string so that the knot appears to have more than 3 crossings. This is an un-simple arrangement, but it remains mathematically/topologically equivalent to the Trefoil knot you began with.

Take a moment to try tying a few more knots from the knot zoo: 3_1, 4_1, 5_1 and 5_2.

Notice the similarity between the 3_1 and 5_1. These knots, and the first kind of all odd-numbered knots, are classified as torus knots. Play around with them and see how different from the original diagram you can get them to look.

The result of most every run in our experiment looked like a jumbled mess of string. A lot of the jumbles were easily simplified into the unknot (0_1), the trefoil (3_1), the figure-8 (4_1), or the 5_1. Some of the jumbles were not so easily simplified, however.

The subtle differences between the sub-types of knots with six or more crossings are difficult to identify, especially in a short time period. To make identifying these knots easier and to assure correct identification, we applied a more sophisticated mathematical analysis developed by Vaughn Jones and Louis Kauffman.

The tool they devised, called the Jones Polynomial, is basically an equation you can calculate for a knot. Every knot in the knot zoo has a unique Jones Polynomial, and each knot's polynomial can be calculated from any representation, even a jumbled-up non-simple representation. We calculated the Jones Polynomial of all the knots produced in our experiment, and from this we could obtain almost all of their minimum crossing numbers. Over the course of three thousand trials, we saw all of the knots in the zoo through the seven crossings and a fair selection in the range between 8 and 11.

The results of these experiments were interesting because they revealed that spontaneous knot formation is not random or unpredictable. Depending on a few physical conditions, string usually has a non-zero chance of becoming a simple knot all the time, especially if it is confined in a bag or box.

Start keeping track of knots you happen upon in your life. After playing with the knot zoo, you will probably start noticing that the spontaneous knots in your life have favorite places to occur.