Knotted Walks

Mountaineers and mariners have long recognized the perverse tendency of loosely coiled ropes to acquire knots. The same sort of tangling can happen to string or thread nestled in a drawer or even to a long hose lying in an untidy heap in the corner of a garden shed.

Given that it normally takes some effort to create a knot, the spontaneous formation of knots in ropes and strings can appear rather puzzling. Having no obvious explanation of the effect, frustrated users can't help but acknowledge this knotting phenomenon as just another manifestation of Murphy's Law: If something can go wrong, it will. In this case, "if a rope can become knotted, it will."

Scientists typically dismiss the attention given to such occurrences as little more than a consequence of selective memory for when things go wrong. However, Robert A.J. Matthews, a science writer and a computer science researcher at Aston University in Birmingham, England, takes a decidedly different view. "The awful truth is that many of the most famous manifestations of Murphy's Law actually do have a basis in fact," he insists.

In situations ranging from the tendency of toast to land butter side down and the prevalence of unmatched socks in drawers, Matthews has applied physical theory and mathematical principles to elucidate the mechanisms underlying the behavior. In many cases, he has found that physics and mathematics conspire to make our lives miserable.

The secret of spontaneously knotting ropes is tied to the mathematics of self-avoiding random walks. "Its roots lie in a recently discovered theorem in topology, found during attempts to solve a puzzle in polymer chemistry," Matthews notes.

Suppose a walker stands at a vertex of a three-dimensional cubic grid. The walker can take a step from one vertex to the next in any one of six directions: left, right, forward, backward, up, or down. Randomly choosing each direction with equal probability (perhaps by using a die), the walker traces a path that wanders from vertex to vertex, sometimes crossing itself and sometimes retracing steps.

Example of a random walk in three dimensions.

The long molecular chain of a polymer floating in a solvent resembles a three-dimensional random walk, with the small molecular units (called monomers) as vertices and the bonds between the units as the steps of the walk. However, to account for the fact that no two monomers can occupy the same region of space, the path has to be modified to one that doesn't intersect itself. The result is a self-avoiding random walk, in which no vertex is visited more than once.

In the early 1960s, researchers studying linear polymers observed that as the length of a molecular chain increases, its likelihood of becoming tangled also grows. If a chemical reaction then bonds the two ends together, the entanglements get permanently locked within the resulting ring. Indeed, it was conjectured that for a sufficiently long chain, the ring polymer would almost certainly contain at least one knot.

In 1988, mathematician De Witt L. Sumners of Florida State University in Tallahassee and chemist Stuart G. Whittington of the University of Toronto proved that conjecture by modeling a tangled polymer chain as a self-avoiding random walk. They also proved a more general result: Nearly all sufficiently long self-avoiding random walks on a simple cubic lattice contain a knotted pattern.

Example of a self-avoiding random walk that traces out a trefoil knot.

In everyday terms, it's possible to interpret the latter result to mean that, if a rope can be regarded as a self-avoiding random walk, all but a very few sufficiently long ropes will contain at least one knot.

"The question is: Can the identification of these everyday objects with [self-avoiding random walks] really be justified?" Matthews asks. If the rope is sufficiently flexible, the answer is yes.

Dumped in a heap in the corner of a garden shed, a rope or hose will undoubtedly be thrown about from time to time, allowing it to explore the three-dimensional space in which it is embedded. "Not a great deal of 'neglect' is needed to meet this requirement, as is shown by putting some light thread or jewellery chain into a bag and briefly jumbling it up," Matthews says. "Carefully extracting the result -- to prevent gravity imposing order on the jumbled thread -- and pulling on the two ends usually reveals a simple trefoil knot." In fact, as the length of thread increases, the jumbling maneuver proves an effective way of forming a knot with practically no effort, he adds.

"An obvious solution to the problem of spontaneous knotting is to coil rope up neatly and bind it to prevent it performing any random walks within its three-dimensional space," Matthews suggests. "This is, of course, the time-honoured approach of mariners and climbers, who take great care over the storage of ropes."

Beating Murphy's Law typically demands a prodigious effort, and even then, something else may go wrong!

References:

Matthews, Robert A.J. 1997. Knotted rope: A topological example of Murphy's Law. Mathematics Today 33(July-August):82-84 (article also available at http://ourworld.compuserve.com/homepages/rajm/knotfull.htm).

______. 1997. The science of Murphy's Law. Scientific American (April):88-91.

______. 1996. Odd socks: A combinatoric example of Murphy's Law. Mathematics Today 32(March-April):39-41.

______. 1995. Tumbling toast, Murphy's Law and the fundamental constants. European Journal of Physics 16(June):172-176.

Peterson, Ivars. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.

Stewart, Ian. 1995. The anthropomurphic principle. Scientific American (December):104-106.

Check out the home page of Robert Matthews at http://www.robertmatthews.org.

Illustrations created using Mathematica 3.0 (http://www.wolfram.com).

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.