The secret to tying knots: After reading this article, you will be invincible

Scientists at MIT have published a "knot-tying secret." It not only tells you how to tie a knot, but also explains why. It is exactly: I am not good at fighting, but you are not good at tying knots.

From ancient times to the present, human beings have experienced the baptism of knots and oracle bones, lead and fire, pen and paper, light and electricity. The Book of Changes records: "In ancient times, people used knots to govern, and later sages changed it to written contracts." This means that our ancestors initially used knots in ropes to record events. The shape and number of knots had a customary connotation, similar to legal provisions. Later, oracle bones, silk, bamboo and wooden tablets were developed, and then paper was developed.

Knots, a kind of entanglement of threads, have always played a seemingly small but very important role in the development of human society. Not only in early records, but also in modern times, climbers, sailors, construction workers, surgeons... all rely on the help of knots, and mothers also rely on "knots" to weave the warmest sweaters and the most beautiful Chinese knots.

But what kind of knot is the strongest? How to tie a knot so that we can safely entrust our lives to it?

01

What determines the strength of a knot?

Previously, the answer often came from a master, a friend, or one's own experience. But is experience always reliable? Is there a scientific rule that can effectively determine whether a knot is strong?

A new study published in the journal Science by Jörn Dunkel and others from the Massachusetts Institute of Technology (MIT) provides a feasible solution to this problem.

The researchers established a suitable physical model by analyzing in detail the subtle interactions between factors such as the shape, elasticity and friction of the knot, and verified it experimentally using a special rope that changes color as the force changes .

A rope that changes color with stress (top) and a simulation result (bottom), which is very consistent | Jörn Dunkel (paper author)

The results show that to determine whether a knot made of two ropes is stable, three factors need to be focused on: the number of intersection points of the ropes, the torsional energy of the ropes, and the tangential force effect of the ropes.

In order to help everyone better understand the key points, scientists gave specific examples and provided simple diagrams. In each case, the stability of each rope was verified by pulling one end and leaving the other end alone. If the rope easily slipped out, it meant that the knot was not stable. The arrows in the diagram indicate the direction in which the rope was pulled.

Now, let's take a look at how these three factors affect the stability of the knot.

1) Number of intersection points of the ropes

Square knot and Alpine bow | Reproduced from Ref. 1

In the simplified line model, we can see that for the common "square knot", the number of intersections between the two ropes is 6. The number of intersections of the Alpine bow knot commonly used by climbers is 12, which is much more than the square knot. Both people's experience and the simulation model and experimental results used in this study have confirmed that the Alpine bow knot with more intersections is much stronger than the square knot.

Of course, in addition to the different number of intersection points, there are also differences in the following factors.

2) Torsional energy of the rope

A rope with low torsional energy (left) and high torsional energy (right) will "spin" like a pair of chopsticks, while a rope with high torsional energy will "twist" like a twisted dough stick | Jörn Dunkel

Take the twisting state of the blue rope as an example. In the left picture, the blue rope has a tendency to rotate counterclockwise (just like the "rotation" of chopsticks when rubbing them) due to the friction forces exerted by the two red ropes in different directions. At this time, the blue rope is easy to rotate but not easy to twist, that is, it has lower torsional energy. The knot has low torsional energy, is easy to rotate and slip, and has poor stability .

If the direction of one of the forces changes, as shown in the right figure, the effect of this force will change to make the blue rope rotate clockwise. In this way, under the combined effect of clockwise and counterclockwise rotation trends, the blue rope is easy to twist (similar to the "twisting" of twisting a twist) but not easy to rotate, and has higher torsional energy. The knot has higher torsional energy, is not easy to rotate and slip, and has better safety .

Thus, in a complex knot, when the combined effect of the directions of the friction forces at the intersections gives the entire knot a higher torsional energy, the knot will be more stable.

Next, we take the square knot and the "granny knot" which is very similar to the square knot as examples to feel the effect of torsional energy on the stability of the knot.

Square knot and granny knot | Reproduced from Ref. 1

From the line model diagram above, we can see that the friction generated at the three intersections in the upper part of the square knot makes the blue rope tend to rotate clockwise, while the three intersections in the lower part make it rotate counterclockwise. For the grandmother knot, the friction at all six intersections makes the blue rope rotate clockwise, and the rope in this case is prone to rotation and slippage. If we use the red rope as the research object, we can get the same result.

Therefore, although both have 6 intersections, due to the difference in torsional energy, the stability of the granny knot is far less than that of the square knot . It is a common mistake made by many novice rock climbers to mistakenly tie a square knot for a granny knot, which may even cause danger. Therefore, the granny knot is also jokingly called the "amateur square knot".

3) Tangential force effect of the rope

The torsional energy mentioned above is the interaction of the vertical forces between the ropes. When the knot is tightened, the ropes are in close contact with each other, and the tangential force generated when they slide relative to each other along the rope direction cannot be ignored.

Consider two parallel ropes in close contact. Compared with pulling in the same direction, the friction will be greater when pulling in the opposite direction, and the knot they form will be more stable. We try to compare the square knot with another similar square knot, the "thief knot". The difference between them is that the two force-applying ends of the square knot are on the same side, while the force-applying ends of the thief knot are on the opposite sides.

The Square Knot and the Thieves Knot (they look identical, but people pull them differently, see the black arrows in the left image) | Reproduced from Ref. 1

From the simplified line model diagram, it can be seen that both the square knot and the thief's knot have 6 intersections and have the same torsional energy. However, experienced climbers will know that the square knot has a better stability than the thief's knot. The difference between them lies in the effect of tangential force.

For the square knot, the rope segments in the middle have opposite pulling directions, and the pulling directions of the upper and lower four pairs of rope segments are also opposite; while for the thief's knot, only the middle rope segment has opposite pulling directions, while the pulling directions of the upper and lower four pairs of rope segments are the same.

Therefore, the effect of the tangential force in the thief's knot is weaker than that in the flat knot, and its firmness is worse . This may be the main reason why the thief's knot is only used by thieves to make marks (the origin of the name "thief's knot") and is not widely adopted by climbers.

02

Small research with big applications

In this way, we analyze the topological structure of the knot (a minimalist structure abstracted from an entity, representing some "essential" characteristics), obtain the three important knot characteristics of the number of rope intersections, torsional energy and tangential force effect, and we can qualitatively predict its stability.

The scientists also found that among all the knots analyzed in this study, simulation and experimental results showed that the Zeppelin Knot was the most stable one and was safer than the Alpine Bow Knot popular in rock climbing .

Super Stable Zeppelin Knot | Reproduced from Ref. 1

This study provides people with a scientific and simple method to predict the stability of a knot by analyzing the shape characteristics of the knot.

But you might think that even without this research, rock climbers, sailors, surgeons, construction workers, and mothers who knit sweaters can obtain excellent knot schemes through long-term practice and experience inheritance. Then you think this research is simple.

This universal rule of judgment is particularly important when scientists study the mechanical properties of microscopic "knot" structures such as DNA (deoxyribonucleic acid), liquid crystals, plasmas, quantum fluids, proteins, polymers, etc. This is also the far-reaching significance of this topological mechanics research on knots and entanglement.

The study of "knots" has broad application prospects. The picture shows DNA under an electron microscope | wikipedia.org

As for most of us, after reading this article, I guess everyone will be a knot master. From now on, you can proudly say to others: I’m not good at fighting, but you are not good at tying knots.

References

[1]VP Patil, JD Sandt, M. Kolle, J. Dunkel, Topological mechanics of knots and tangles, Science 367, 71–75 (2020).

[2]https://phys.org/news/2020-01-mathematical-stability.html

[3]https://www.npr.org/2020/01/02/793050811/a-knotty-problem-solved

[4]http://news.mit.edu/2020/model-how-strong-knot-0102

[5]https://www.sciencenews.org/article/color-changing-fibers-mysteries-math-physics-how-knots-work

[6]https://www.scientificamerican.com/article/color-changing-fibers-unravel-a-knotty-mystery/

Author | Long Hao PhD in Microelectronics and Solid-State Electronics

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