How can knots be accurately modelled?

Jon Lawson

In sailing, rock climbing, construction and any activity requiring the securing of ropes, certain knots are known to be stronger than others. Any seasoned sailor knows, for instance, that one type of knot will secure a sheet to a headsail, while another is better for hitching a boat to a piling. 

But what exactly makes one knot more stable than another has not been well-understood, until now. 

MIT mathematicians and engineers have developed a mathematical model that predicts how stable a knot is, based on several key properties, including the number of crossings involved and the direction in which the rope segments twist as the knot is pulled tight. 
“These subtle differences between knots critically determine whether a knot is strong or not,” said Jörn Dunkel, associate professor of mathematics at MIT. “With this model, you should be able to look at two knots that are almost identical, and be able to say which is the better one.”

“Empirical knowledge refined over centuries has crystallised out what the best knots are,” added Mathias Kolle, the Rockwell International Career Development Associate Professor at MIT. “And now the model shows why.”

Why colour is important for pressure analysis

In 2018, Kolle’s group engineered stretchable fibres that change colour in response to strain or pressure. The researchers showed that when they pulled on a fibre, its hue changed from one colour of the rainbow to another, particularly in areas that experienced the greatest stress or pressure. 

Kolle was invited by MIT’s maths department to give a talk on the fibres. Dunkel was in the audience and began to cook up an idea: What if the pressure-sensing fibres could be used to study the stability in knots? 

Mathematicians have long been intrigued by knots, so much so that physical knots have inspired an entire subfield of topology known as knot theory — the study of theoretical knots whose ends, unlike actual knots, are joined to form a continuous pattern. In knot theory, mathematicians seek to describe a knot in mathematical terms, along with all the ways that it can be twisted or deformed while still retaining its topology, or general geometry. 

“In mathematical knot theory, you throw everything out that’s related to mechanics,” Dunkel said. “You don’t care about whether you have a stiff versus soft fibre — it’s the same knot from a mathematician’s point of view. But we wanted to see if we could add something to the mathematical modelling of knots that accounts for their mechanical properties, to be able to say why one knot is stronger than another.” 

The physics of Spaghetti

Dunkel and Kolle teamed up to identify what determines a knot’s stability. The team first used Kolle’s fibres to tie a variety of knots, including the trefoil and figure-eight knots — configurations that were familiar to Kolle, who is an avid sailor, and to rock-climbing members of Dunkel’s group. They photographed each fibre, noting where and when the fibre changed colour, along with the force that was applied to the fibre as it was pulled tight.

The researchers used the data from these experiments to calibrate a model that Dunkel’s group previously implemented to describe another type of fibre: spaghetti. In that model, Patil and Dunkel described the behaviour of spaghetti and other flexible, rope-like structures by treating each strand as a chain of small, discrete, spring-connected beads. The way each spring bends and deforms can be calculated based on the force that is applied to each individual spring. 

Kolle’s student Joseph Sandt had previously drawn up a colour map based on experiments with the fibres, which correlates a fibre’s colour with a given pressure applied to that fibre. Patil and Dunkel incorporated this colour map into their spaghetti model, then used the model to simulate the same knots that the researchers had tied physically using the fibres. When they compared the knots in the experiments with those in the simulations, they found the pattern of colours in both were virtually the same — a sign that the model was accurately simulating the distribution of stress in knots. 

With confidence in their model, Patil then simulated more complicated knots, taking note of which knots experienced more pressure and were therefore stronger than other knots. Once they categorised knots based on their relative strength, Patil and Dunkel looked for an explanation for why certain knots were stronger than others. To do this, they drew up simple diagrams for the well-known granny, reef, thief, and grief knots, along with more complicated ones, such as the carrick, zeppelin, and Alpine butterfly.

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