Random Knots

By Ivars Peterson

Muse, March 2005, p. 29.

Have you ever left a necklace or a piece of string lying around on a table in a jumbled heap? There's a good chance that it will have formed a knot when you pick it up again, especially if it has been jostled a little. The same thing can happen to a garden hose left in an untidy pile on the ground.

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

Topologists—mathematicians who study shapes—have investigated how knots can form accidentally. Imagine a three-dimensional grid made up of lines that define a set of evenly spaced rows, columns, and stacks. Suppose a "walker" were to start at one point, or vertex, of this grid. The walker steps randomly from one vertex to the next vertex in any one of the six directions available from a given point.

A self-avoiding walk in three dimensions can create a knotted path.

Because the path is chosen randomly, perhaps by rolling a die to determine the direction of each step, topologists call the walker's path a random walk. When the walker is not allowed to revisit the same vertex a second time, the path is called a self-avoiding random walk. Mathematicians have proved that the longer the random walk, the greater its chance of forming a knot.

So, it shouldn't really come as a surprise that a rope, necklace, or garden hose—if it's long enough—is quite likely to settle into a knot. There's no getting away from knotty situations!

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