A Theory of Knots

[SailOar]

A new mathematical model predicts a knot's stability


MIT mathematicians and engineers have developed a mathematical model that predicts how stable a knot is, based on several key propectyrties, 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," says 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."....

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

In comparing the diagrams of knots of various strengths, the researchers were able to identify general "counting rules," or characteristics that determine a knot's stability. Basically, a knot is stronger if it has more strand crossings, as well as more "twist fluctuations"—changes in the direction of rotation from one strand segment to another.

For instance, if a fiber segment is rotated to the left at one crossing and rotated to the right at a neighboring crossing as a knot is pulled tight, this creates a twist fluctuation and thus opposing friction, which adds stability to a knot. If, however, the segment is rotated in the same direction at two neighboring crossing, there is no twist fluctuation, and the strand is more likely to rotate and slip, producing a weaker knot.

They also found that a knot can be made stronger if it has more "circulations," which they define as a region in a knot where two parallel strands loop against each other in opposite directions, like a circular flow.

By taking into account these simple counting rules, the team was able to explain why a reef knot, for instance, is stronger than a granny knot. While the two are almost identical, the reef knot has a higher number of twist fluctuations, making it a more stable configuration. Likewise, the zeppelin knot, because of its slightly higher circulations and twist fluctuations, is stronger, though possibly harder to untie, than the Alpine butterfly—a knot that is commonly used in climbing. ..... [???]

Topological mechanics of knots and tangles, Science (2020)
http://dx.doi.org/10.1126/science.aaz0135

[Dockhand]

Did it take into account which knot makes the rode weaker in strength? Which the knot may hold but break sooner than a different knot.

[sailormed]

That is the subject of another study. After all as a science guy you are judged on the number of publications of your articles. So better split up your subjects.

[Dr. D]

Cool! Thanks for posting.

[Labezin]

looks like interesting but unfortunately they ask to pay something to accesss the report. Not sure its worth it ?

[Mike Banks]

The breaking strength reduction of any knot is REALLY important--and the size of the completed knot is also important, but knots work by using friction, and the more friction one can get for the least bending radius of the cable and parts in tension, the stronger the knot will be. Size is convenience and aesthetics are a consideration.

A bowline is one of the commonest and most useful of all knots used aboard and ashore--but it has a large reduction factor in the strength of the cable. That does not matter if, as is usually the case, the cables used are much stronger than they need to be in order to get a bigger diameter for ease of handling.

It is not my go-to knot in any but emergency situations. A bowline is about the quickest loop knot one can tie. That gives it a big advantage--but where there is time to spare, I use a blood loop, preferably with a double turn around the shackle or ring, and if it is to be a permanent attachment I like to use a splice and thimble, but if time is short, a properly made and tightened down Blood loop knot snugs down very nicely over a thimble.


A sailor from port of Rabaul--
Wrote of knots on his cabin wall-
Said "The strength I've incurred
When expressed as a surd--
Is five times the square roof of f*** all."

Some quick knots are not always safe for use in synthetic materials.

[tarian]

I think it is amazing that we know need mathematicians to tells us what sailors have known for thousands of years and that is what knot is best used for a certain situation on a boat , modelling can only go so far and then you have to field test the product , seems like this boat has already sailed

[CDR Smith]

"propectyrties" What an awesome word!
(For goodness sake, don't edit the original post!)

[JoeRobertJr]

Originally Posted by tarian

I think it is amazing that we know need mathematicians to tells us what sailors have known for thousands of years and that is what knot is best used for a certain situation on a boat , modelling can only go so far and then you have to field test the product , seems like this boat has already sailed

Exactly. Or to say it another way, if the theory suggests that Knot B is better in a particular application, but nautical tradition has favored Knot A, then it would take some demonstration of Knot B's superiority, and Knot B would probably have to be vastly superior to become more popular.

On the other hand, it is probably the case that a particular situation already has a few knots that are popular. Some sailors like Knot A, others Knot B, which makes for good pub arguments. Now, the sailor with the MIT report can win the argument!

I am anxiously awaiting MIT's development of a theoretical model for best anchor.

[tarian]

Originally Posted by JoeRobertJr

Exactly. Or to say it another way, if the theory suggests that Knot B is better in a particular application, but nautical tradition has favored Knot A, then it would take some demonstration of Knot B's superiority, and Knot B would probably have to be vastly superior to become more popular.

On the other hand, it is probably the case that a particular situation already has a few knots that are popular. Some sailors like Knot A, others Knot B, which makes for good pub arguments. Now, the sailor with the MIT report can win the argument!

I am anxiously awaiting MIT's development of a theoretical model for best anchor.