Unravelling a knotty mystery

How can mathematical theory predict the strongest knots in real life? The answer to that question has remained elusive despite humans relying on knots for thousands of years while fishing, hunting, building shelters and harnessing animal power. By using colour-changing fibres to experimentally test a mathematical theory, researchers have begun developing more realistic models that can elucidate the rules governing stability in knots.

The mathematical theory of knots has typically focused on classifying their different entangled structures without accounting for mechanical stress and strain. To better predict knots’ strength, Massachusetts Institute of Technology (MIT) researchers experimented with those made from special photonic fibres that change colour as they stretch.

Doing so allowed the team to compare the mechanical-strain predictions of their mathematical models against those of the colour-changing fibre experiments and therefore develop more sophisticated models that can simulate the intricacies of knots and, perhaps, more complex tangled structures.

“My collaborators developed beautiful fibres that change colours as you stretch them,” says Jörn Dunkel, a mathematician at MIT. “They offered, for the first time, the ability to see in knots where the stress and strain goes in the fibres.”

Such work allowed Dunkel and his colleagues to come up with three counting rules for predicting knot stability, as detailed in their study, published in Science. They focused on analysing the strength of different “bend” knots commonly used by sailors and climbers to tie two pieces of rope together.

The first rule suggests that knots gain stability when the two strands have a higher number of crossing points where they come in contact. The second one is that knots become more stable if strand segments at neighbouring crossing points rotate in different directions and create opposing friction. And the third rule is that knots gain stability when strands slide tangentially against each other to create friction while being pulled in opposite directions.

Implications

Such findings have some practical implications. For example, the study suggests that a type of bend knot called the Zeppelin knot offers greater stability than the more popular alpine butterfly knot. “The nice thing is that with these basic rules, you can get quick intuition about which knots are more stable or less stable,” Dunkel explains. “We can also use them as a starting point to explore more accurate models.”

The MIT team’s work would not have been possible without the colour-changing photonic fibres that were originally developed back in 2013. Such photonic fibres have a coating made from a periodic arrangement of two different types of elastic rubber, each with different material properties affecting their interaction with light. As a photonic fibre stretches under strain, its original colour changes because the coating gets thinner and alters the original optical properties. Such fibres are examples of the same principle behind the rainbow of colours found in soap bubbles, butterflies and beetles.

“If, for instance, a relaxed fibre is red, it will change colour through orange, yellow, green and blue to purple as it is stretched more and more,” says Mathias Kolle, a mechanical engineer at MIT and a co-author of the study. “In addition, we can tailor the spectral landscape of the fibres and play different tricks to generate other colour progressions.”

The experiments with the colour-changing photonic fibres provided the reality check so that the mathematical models incorporating the same data could accurately predict the strongest types of knots. After simulating the strongest knots, researchers drew flat knot diagrams to examine the pattern of strands that make up each one. They also tested the real-world strength of knots by tying two pieces of Dyneema synthetic fibre together, hanging one end of the combined “rope” from a gripping mechanism and then attaching progressively heavier weights to the rope’s dangling end.

The result is “a very interesting blend of experimental work and qualitative theoretical work,” says Louis Kauffman, a professor emeritus in mathematics at the University of Illinois at Chicago, who was not involved in the MIT study.

Promising though they may be, such methods are not without limitations. The flat-knot diagrams seem to do well in modelling real-world knot behaviour so far, but more complex tangles may require 3-D modelling to fully account for how all the different forces are interacting. “This raises the very fascinating question of generalising these models into full three-dimensional reality,” Kauffman says.

Another limitation, Dunkel says, is that the mathematical models do not account for how different fibre materials affect the friction within a knot. Still, the team is eager to see if its models can find similar rules in systems beyond knots of fibre and string. For example, other research groups have discovered knot structures appearing within the liquid crystals used in technologies such as flat-screen TVs.

Separately, Kolle and his engineering team are continuing to develop their colour-changing photonic fibres with the goal of making them more efficient in larger and more durable forms. So far, the researchers can make fibres about 15 cm in length in the lab, with each capable of thousands of stretching, colour-changing cycles that traverse the entire visible spectrum.

The engineers have their eye on eventually making the fibres suitable for applications such as colour-changing sportswear and fashion items, where alterations in colour might help signal how well the fabric is holding up over time. Weaving such fibres into medical bandages might also help physicians and ordinary people visually check where bandages are applying maximum pressure to correctly staunch bleeding.

But commercialisation possibilities aside, Kolle remains thrilled that the colour-changing photonic fibres could help his colleagues
untie a rather difficult knot of a problem. “It was very easy to get excited about it—the colour-changing fibres being useful for really creative fundamental research,” he says.