By Vishal Patil and Jörn Dunkel
One can trace knot theory in mathematics and theoretical physics back to Lord Kelvin’s 19th century description of atoms as knotted vortices in the ether. However, knots have been a staple of the real world for thousands of years — from fastening ancient tools to securing cargo bound for Mars. This long history, coupled with knots’ importance in activities ranging from sailing and climbing to surgical procedures, has resulted in a wealth of empirical knowledge about their mechanical properties. For example, climbers know that the knot they tie to rappel down a cliff is not the same as the knot that binds two ropes together. Yet researchers know comparatively little about the mechanics of knots from a theoretical perspective.
Predicting the behavior of knots from a diagram is particularly challenging, as similar-looking knots can have strikingly different mechanical properties. A famous example of this concerns the difference between the reef knot and granny knot (see Figure 1). Although these two knots are frequently mistaken, the granny knot slips and unravels easily when pulled, whereas the reef knot is more secure. This is of practical importance, as most of us use one of these two knots to tie our shoelaces. The fact that small changes can yield significant physical consequences suggests that knot strength has a topological component. Generalizing the reef and granny knot problem led us to consider the classic “prisoner’s escape” problem: what is the most secure way to tie two ropes together? In our work, which was published in Science this January, we address this question by exploring the way in which certain topological counting rules interact with mechanics and friction.
Figure 1. The granny knot (left) and reef knot (right). Arrows indicate pulling direction, with the other ends moving freely.
To quantify knot security, we first built a simulation framework that combined the Kirchhoff equations for elastic fibers with contact conditions to model friction. A key challenge in modelling knots is accurately reproducing their stress distribution. We were able to validate our simulations by comparing them to newly developed color-changing fibers (see Figure 2). These fibers consist of layers of cladding around a central core. When placed under strain, the refraction of light as it travels through the layers causes the fiber to change color. This setup allowed us to visualize the stresses within the knot in real time.
Figure 2. Experiment (left) and simulation (right) indicate the stress in a tight trefoil knot. The experiments were performed by Joseph Sandt in Mathias Kolle’s lab at the Massachusetts Institute of Technology.
Using our validated numerical model, we identified three topological parameters that govern knot behavior: twist friction, circulation friction, and crossing number. One can calculate these parameters—which are derived from the frictional contact interactions within the knot body—via a diagram of the knot in its loose, elastically relaxed state. When two strands of the knot are dragged past each other, they interact via a force and a torque. Think of the torque as each strand imparting a twist on the other (see Figure 3a). These twists correspond to crossings in the knot diagram (see Figure 3b). If all twists are in the same direction, the knot can roll free — as with the granny knot. On the other hand, if some twists are in opposite directions—as with the reef knot—the knot will lock as it is tightened. This conception of twist friction is closely related to a mathematical knot parameter known as writhe, thus illustrating the connection between knot mechanics and topology.
Together with twist friction, we also found a circulation friction that arose from an in-plane, self-contact interaction within the knot. When two strands pass over each other, they are either moving in the same direction or opposite directions. The former is a low-friction interaction, while the latter is high friction. One can think of circulation friction as arising from the faces of the knot diagram (see Figure 3b). Our final parameter—the diagram’s crossing number—captures the fact that more complex knots are generally more secure due to increased friction from self-contact.
Figure 3. 3a. Twists of the same handedness (left) cause rolling, while opposite twists (right) are higher energy. 3b. Twist friction arises from signed twists at crossings, while circulation friction arises from circulations at faces. Pulling direction provides curve orientations. The granny and reef knots have different twist frictions but the same circulation friction.
To verify this topological model, we simulated 16 different knots and demonstrated the way in which their security is predicted by a phase diagram with axes that represent twist friction, circulation friction, and crossing number. We also tested six of these knots experimentally by mimicking a “prison escape” style experiment. The setup consisted of two strands of Dyneema fiber tied together; one end was clamped while the other was loaded with weights. Knots are a very noisy system, with dynamics strongly dependent upon complex surface interactions. Nevertheless, our experiments indicated good qualitative agreements with both the simulation framework and our topological model.
Researchers can utilize our topological model to understand the security of a class of knots. The aforementioned parameters could also have applications to continuum systems beyond knotted ropes, such as tangles of vortices in complex fluids or tangles of proteins and DNA. When it comes to understanding physical knots, however, nontopological variables—like surface contact geometry—also play an important role. A combination of topological arguments and more refined models (which account for these effects) can provide a framework for understanding and developing new classes of knots.
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Vishal Patil is a graduate student in the Department of Mathematics at the Massachusetts Institute of Technology. |
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Jörn Dunkel is an associate professor in the Department of Mathematics at the Massachusetts Institute of Technology. |