The Mathematics of a Well-Tied Knot

Fibers that change color under pressure helped researchers predict knot performance

Color-Changing Fiber
MIT researchers develop a mathematical model to predict a knot’s stability with the help of color-changing fibers. Photo by Joseph Sandt

Knots are some of the oldest and most-used technologies that humanity employs. But knowledge of different knots—their strengths, weaknesses and best applications—has generally come from practical experience. Now, a team of mathematicians and engineers at MIT has combined theoretical and experimental research to explain the math and physics behind popular knots’ stability.

The new study, published last week in the journal Science, paired mathematical knot theory with a color-changing fiber developed in 2013. Because the fiber changes color under pressure, the researchers were able to measure physical properties and add data to their computational knot models. They came up with three rules that determine a knot’s stability.

The improved model allowed the researchers to untangle the reasons that similar-looking knots behave very differently when pulled. Speaking with NPR’s Nell Greenfieldboyce, mathematician Vishal Patil gives the example of the granny knot and the reef knot, both of which loop two ropes together but differ by one overlap.

“If you pull on the reef knot, it tends to hold,” Patil tells Greenfieldboyce. “And if you pull on the granny knot, it tends to slip quite easily. The fact that they behave so differently suggests that there must be some story there, something you can say mathematically and physically about them.”

The team began by using the color-changing fiber, which co-author Mathias Kolle helped develop, to tie a few simple knots. The fiber turned green and yellow under high pressure and remained red or orange without stress. The data collected in these experiments was then integrated into the calculations of a computer model of ropes and knots.

After confirming the colors in photos of the experiment matched the pressures shown in computer models of the same knots, the team modeled a series of more complicated rope configurations. Per Scientific American’s Jeremy Hsu, the researchers focused on “bend” knots, used by sailors and climbers to fasten two pieces of rope together. Incidentally, Kolle is an avid sailor, and other members of the team enjoy rock climbing.

Knots that withstand the most strain are the strongest, and those that withstand the least are the weakest. By studying and ranking seven knots, the researchers identified three characteristics that allow a knot to put up with more strain.

First, knots are more stable with each additional crossing point, where one length of rope comes in contact with another. Next, if strands at neighboring crossing points rotate in opposite directions, it will create opposing friction and also increase stability. Friction from strands sliding against each other in opposite directions provides the final contribution.

The study is “a very interesting blend of experimental work and qualitative theoretical work,” mathematician and knot theory specialist Louis Kauffman, who was not involved in the paper, tells Hsu.

The research allowed the team to identify the reason the reef and granny knot withstand different amounts of strain—the reef knot has more twists, increasing friction and making it more stable. In the future, this type of research could be used to choose or create the right knot for any application.

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