For the first time, scientists have predicted how the force needed to pull a knot tight depends on shape of the knot. The advance could lead to a more precise way of tailoring specific knots to particular needs such as surgical operations and might also improve our understanding of naturally occurring knots in DNA and elsewhere.
As any sailor or climber knows, the strength of a knot depends very much on the knot's configuration. For example, suppose you want to tie a sturdy reef knot (also known as a square not) in your shoelaces. Start by crossing the lace in your right hand in front of the one in your left hand and then threading the right hand one over and under the left to form a simple "overhand" knot. Once you've swapped hands and pulled this snug, you repeat the process to make a second overhand knot on top of the first—except you have to cross the end that's now in your right hand behind the one in your left (adding loops to make the knot easier to untie). If you just repeat exactly what you did before, you'll end up with a weaker granny knot.
Audoly's theory proved successful up to a point, being able to predict the correct force for overhand knots with either one or two twists. However, experiments carried out by mechanical engineer Pedro Reis and colleagues at the Massachusetts Institute of Technology (MIT) in Cambridge showed that the theory came unstuck for greater numbers of twists. Those experiments involved tying knots with varying amounts of twist in a highly elastic nickel titanium wire, clamping that wire to a table, and then pulling the knots tight with a mechanical arm. By measuring the force applied to the arm, Reis and co-workers found that a knot with 10 twists needed a tug about 1000 times stronger than one with a single twist, which Audoly had failed to predict.
In the new study, Audoly and the MIT researchers have gotten together to produce a theory consistent with the experimental results. They modeled the forces that exist inside the stretched wire, assigning a greater role to friction than did the earlier theory—given that more twists means greater rubbing between the two ends of the string. They then wrote down a formula for the pulling force in terms of three (known) variables: the thickness of the wire, its stiffness, and the number of twists. Plotting a graph with suitable combinations of these variables on the two axes, the researchers traced a straight line that coincided almost perfectly with the experimental data points. Extrapolating the line should generate reliable force values for knots with more than 10 twists, the team reports online before print in Physical Review Letters.
The theory only applies to overhand knots, Reis points out, and is certainly not "a grand unified theory of knots." Nevertheless, he hopes that it can serve as a stepping stone to theories describing more complex knots, and that in the future it might allow surgeons, for example, to tune the strength of a knot by altering its number of twists. "A lot of knowledge about knots is empirical," he says. "We have taken a more rational approach and have created a predictive framework. It is what the community was lacking."
Knot experts agree that the latest research could have practical benefits. The experimental measurements "very nicely demonstrate" the predictive power of the model, says Doug Smith, a physicist at the University of California, San Diego. Potential applications could include tuneable shock absorbers or better exercise stretch bands, he says. In the latter case, he says, knotted cord could have advantages over existing rubber bands because their stretchiness could be tailored by changing the number or type of knots.
Knots occur naturally in many molecular systems such as folding proteins and DNA, notes Louis Kauffman, a mathematician at the University of Illinois, Chicago. He says that being able to calculate the strength of the forces holding the knots together would "help in understanding how cells divide and how the processes involving RNA and DNA work."