Simpler Brunnian "rubberband" loops?

[egmoen]

TL;DR

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.

 

[egmoen]

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?

 

[jim mcnamara]

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

 

[egmoen]

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?

 

[jim mcnamara]

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.