The Mathematical Tourist
By Ivars Peterson
October 1, 2007
Coil a long rope into a box, then shake the box. It's very likely that the rope will emerge bearing a knot.
The same perverse tendency of spontaneous knot formation also plagues necklaces loosely stored in drawers or headphone cords stuffed into pockets or backpacks.
A coiled headphone cord, once unraveled, is likely to contain a knot.
Now, physicists have investigated this pervasive phenomenon, performing experiments to identify the physical factors that lead to spontaneous knot formation and applying mathematical knot theory to analyze the resulting knots. Dorian M. Raymer and Douglas E. Smith of the University of California, San Diego describe their findings in a paper to be published in the Proceedings of the National Academy of Sciences.
The researchers discovered that, when a string is tumbled inside a box, complex knots often form within secondsif the string is long and flexible enough. For example, for a string of diameter 3.2 millimeters, coiled inside a cubic box 0.3 meter wide, which was spun at 1 revolution per second for 10 seconds, no knots formed when the string was shorter than 0.46 meter. When a tumbled string had a length between 0.46 and 1.5 meter, the probability of knot formation went up sharply.
This illustration represents the knot experiment, in which knots form in a tumbled string. Dorian Raymer, UCSD.
Overall, the experiments suggest that the probability of knot formation increases to nearly 100 percent for long agitation times, long length, and high flexibility.
To determine what knots had formed after a string was tumbled, the physicists lifted the ends of the string directly upward from the box, then joined the ends to form a closed loop. They photographed the loop, creating a two-dimensional knot diagram showing where the loop crossed over or under itself. The researchers then applied mathematical knot theory—calculating each tangle's Jones polynomial—to identify the knot.
The analysis produced a striking result. In 3,415 trials, Raymer and Smith observed knot formation 1,127 times, identifying 120 different types of knots, having a minimum crossing number up to 11. Moreover, the researchers found instances of every prime knot with up to seven crossings. The simplest possible knot—a trefoil knot—has three crossings. Composite knots—for example, a pair of trefoil knots—formed far less frequently than did prime knots.
Digital photos of knots are combined here with computer-generated drawings based on mathematical calculations. Dorian Raymer, UCSD.
The fact that the majority of the observed knots were prime suggests that knotting tends to occur from one end. Once a knot starts at one end, it's highly unlikely (though occasionally still possible) for a knot to start at the other end. Indeed, knot theory dictates that, if a separate knot is formed at each end of a string, the knots can slide together at the center of the string but can't merge to form a single prime knot.
The researchers also developed a simple model to describe knot formation on the basis of random "braid moves" of a string's free end. With multiple parallel strands lying near a string end, knots form when the end weaves under and over adjacent segments. In effect, the string end traces a path that corresponds to the knot topology.
According to a basic theorem of knot theory, such braid moves can generate all possible prime knots. This is consistent with the observation that the physicists found all prime knots with up to seven crossings among their tumbled strings.
So, what are the chances of ending up with a knot when you unravel your carefully coiled headphone cord? They're annoyingly high, especially if the cord is long, thin, and flexible.
Comments are welcome. You can reach Ivars Peterson at firstname.lastname@example.org.
2007. Unraveled: The mystery of why strings tangle. New Scientist (Sept. 30).
Peterson, I. 1997. Knotted walks. MAA Online (Nov. 3).
Raymer, D.M., and D.E. Smith. In press. Spontaneous knotting of an agitated string. Proceedings of the National Academy of Sciences.Seethaler, S. 2007. UC San Diego physicists tackle knotty puzzle. UCSD News Center (Oct. 1).
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