Random Walks, Random Knots

21. Galactic Gridlock

Random Walks, Random Knots

You're done with your portable music player, so you carefully coil the headphone cord around the player and stuff it in your pocket. The next time you take it out, however, you find that you have to unravel the cord and undo a knot before you can go back to listening to your music.

A coiled headphone cord, once unraveled, is likely to contain a knot.

In fact, it's pretty amazing how easily knots can form by themselves—not only in headphone cords, but also in necklaces, coiled ropes, strings of holiday lights, hanks of yarn, or garden hoses.

Sailors and rock climbers know all about that problem, so they take great care to store their ropes in ways that prevent accidental knotting. Because we are used to making an effort to tie a knot, the unintended formation of knots in ropes, hoses, strings, necklaces, and headphone cords can be both frustrating and puzzling.

Mathematicians have investigated the spontaneous formation of knots using a three-dimensional grid that resembles the Cubic Grid Galaxy (see "Galactic Gridlock"). They imagine a walker standing at one point, or vertex, of the cubic grid. The walker steps randomly from one vertex to the next in any one of six directions available from a given point.

Since the path is chosen randomly, perhaps by rolling a six-sided die to determine the direction of each step, mathematicians call the walker's path a random walk. When the walker is not allowed to visit a vertex for a second time, the path is called a self-avoiding random walk.

Example of a short, self-avoiding random walk in three dimensions.

Mathematicians and scientists use random walks as models for explaining a variety of natural phenomena, including the shapes and folds of polymers, the chain-like molecules that make up plants and animals. Plant and animal by-products such as wood, petroleum, and plastics are also made of polymers. The DNA molecules in the cells that make up our own bodies are polymers, too.

Polymers consist of long chains of "monomer" units. The chains twist and turn and cross over their own paths like a tangled string of beads. Each bead on a string, and each monomer in a polymer chain, is like one step in a random walk.

In fact, polymer chains or strings of beads are like self-avoiding random walks. No two beads can occupy the same place on a necklace, and neither can two monomers on a polymer chain.

In 1988, researchers used a self-avoiding random walk as a model for a polymer chain. They proved that the longer the random walk, or chain, the greater the chance of forming a knot.

Similarly, if you zoom to enough stars in the Cubic Grid Galaxy, your route is almost certain to form a knot.

Knotty Cords

A while ago, two physicists decided to do some experiments to try to find out why knots form so easily in coiled strings. They looked at what happens when a long string is coiled into a box and the box is then tumbled.

The researchers, Dorian M. Raymer and Douglas E. Smith, found that complex knots form within seconds—if the string is long and flexible enough. So, for a given stiffness, a string has to be a certain length before a knot will form. Moreover, the longer and more flexible the string, the better a chance it has of becoming knotted, especially when the string is tumbled or shaken for a long time. Video.

This illustration represents the knot experiment, in which knots form in a tumbled string. Dorian Raymer, UCSD.

The researchers did the experiment over and over again: 3,415 times in all. Knots formed in the strings about one-third of the time. The biggest surprise came when the physicists used mathematics to identify the types of knots that formed in the strings.

Mathematicians called knot theorists have described many different kinds of knots, according to features such as crossings (the number of times a string crosses over itself). The physicists found that their real-life model produced 120 different knots. In fact, their strings formed all of the possible kinds of knots with up to seven crossings—and seven crossings is quite a tangle.

Digital photos of knots are combined here with computer-generated drawings based on mathematical calculations. Dorian Raymer, UCSD.

Raymer and Smith say that knotting tends to start from one end of a string. As a string tumbles, this free end braids itself with strands lying next to it, weaving over and under adjacent segments to create a knot. The more braiding, the more complex the knot.

So, if your headphone cord is long, thin, and flexible (if?!), the chances of it becoming knotted are annoyingly high. Sigh.

NEXT: Walking a Line