Simpler Brunnian "rubberband" loops?

In summary, the conversation discusses the concept of Brunnian "rubberband" loops, which are a series of unknots connected together in a U-shape with their ends looped around the next unknot. This requires 8 crossings. However, the conversation also mentions a simpler method using lark's-head knots, which only requires 6 crossings per pair and allows for easier separation of the structure with a radial cut. It is suggested that this method should be called "an exaltation of larks" and there is some uncertainty about whether it is a unique Brunnian knot. The website knotplot.com is also mentioned as a tool for exploring knot construction.
  • #1
egmoen
3
1
TL;DR Summary
Depictions of Brunnian "rubberband" loops show inividual elements being joined in a way that requires 8 crossings per pair; it seems like if we use lark's-head knots we reduce this to 6.
The standard configuration of Brunnian "rubberband" loops shows a series of unknots each bent into a U-shape, with their ends looped around the middle of the next unknot. (See for instance http://katlas.math.toronto.edu/wiki/"Rubberband"_Brunnian_Links). This connection requires 8 crossings.

If we connect the unknots together using a simpler lark's-head (cow hitch) knot, we still get a set of Brunnian links, since removing any element causes the entire structure to fall apart. But this is much simpler than the method shown above. It requires only 6 crossings per pair, and means that a radial cut through the overall structure only needs to sever two bights, not four. (A picture of a non-Brunnian chain using lark's-head knots can be found here: https://www.cs.bham.ac.uk/research/projects/cogaff/misc/rubber-bands.html)

Since I can't find an earlier description of this possibility, I'm worried that I have missed something that might disqualify this approach.
 
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  • #2
Here's what I'm visualizing. If this works and doesn't have a name yet, it should definitely be called "an exaltation of larks", right?
exaltation.png
 
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Likes jim mcnamara
  • #3
I do not have good answer, but this is both fun and interesting.

This is the only compendium I can see, likely due to paywalls.

http://katlas.org/wiki/Brunnian_link
 
  • #4
Yeah, I saw the katlas page...not much on there.

But would you agree that the loop above can't be separated without cutting an element, and falls apart if you cut any element?
 
  • #5
So it seems. But as to being a unique Brunnian knot, I do not know. These constructs are sometimes called "rubberband" knots. Make one for yourself and play with it. Maybe one of the mathematicians here knows something more substantial.

@mathwonk @lavinia @fresh_42 seem like possibilities.

I think you may have already found https://www.knotplot.com/ which let's you play around with knot construction.
 

1. What is a Simpler Brunnian "rubberband" loop?

A Simpler Brunnian "rubberband" loop is a type of mathematical knot that consists of three or more loops of rubber bands that are intertwined and cannot be separated without cutting the bands. It is named after Hermann Brunn, a German mathematician who first studied these types of knots.

2. How is a Simpler Brunnian "rubberband" loop different from a regular Brunnian "rubberband" loop?

A Simpler Brunnian "rubberband" loop is a type of Brunnian knot that is made up of only three loops, while a regular Brunnian "rubberband" loop can have any number of loops. Additionally, in a Simpler Brunnian "rubberband" loop, all of the loops are connected to each other, while in a regular Brunnian "rubberband" loop, some of the loops may be connected to each other and others may not be.

3. What are some real-world applications of Simpler Brunnian "rubberband" loops?

Simpler Brunnian "rubberband" loops have been used in various fields, such as chemistry, biology, and physics, to model and study complex molecular structures, DNA, and protein folding. They have also been used in engineering to create strong and flexible structures, such as bridges and buildings.

4. Can Simpler Brunnian "rubberband" loops be untangled?

No, Simpler Brunnian "rubberband" loops cannot be untangled without cutting the rubber bands. This is because the loops are intertwined in a way that none of them can be removed without affecting the others.

5. How are Simpler Brunnian "rubberband" loops studied and analyzed?

Mathematicians and scientists use various methods, such as knot theory, topology, and graph theory, to study and analyze Simpler Brunnian "rubberband" loops. Computer simulations and physical models are also used to better understand the properties and behaviors of these knots.

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