advice on knots and ropes

The noise of chains would be annoying with so much motion (as you note), and chains would be more likely to injure someone from an incidental hit or get tangled in long hair.

Obviously I am only talking about the upper connective section of the apparatus.

If additional connective material is required for the feet - then yes, sewn webbing with adjuster buckles would be ideal because obviously, every person is a different height and has different length limbs.

However, the upper connective portion can be swaged wire or chain. This section would not need to be adjustable - its the lower section for the limbs (feet) that should be adjustable.
NOTE: Can also use rated webbing with sewn terminations (terminations that form a connective ‘eye’).

The comment re hair entanglement is unlikely - given that the chains or swaged wire exists above the horizontal pipe/bar which is gripped by the gymnasts hands. Also, the original OP has not given much detail as to precisely what the apparatus will be used for (or the manner in which it will be used). The original post was largely focused on the upper connective section with respect to the swivel. I would also comment that gymnast safety rules dictate that long hair must be tied back/secured. A gymnast would not be allowed on an apparatus with long hair freely waving/flapping about - further reducing the likelihood of hair entrapment.

S-t-r-e-t-c-h will be an issue with any synthetic connective materials in the upper section - due to the force induced by the included angle. Newtons laws of physics means that increasing included angles = higher forces. The ‘included angle’ is the internal angle created at the ‘apex’ point where the swivel connector is located. It is not difficult and inexpensive to get cable cut and swaged to measure. I originally suggested chain because it is easy to cut and has ready made links. Cable will require swaging - but this is inexpensive.

To be honest, I am not 100% clear as to precisely how the apparatus is intended to be used - so my recommendations are based on educated guesses. I would comment though that safety should be paramount - and hand tied knots should be avoided where practicable. The OP is not an engineer or a knot tying specialist so this is another complicating factor. Also, it is conceivable that children may be involved…and if this is true, if it were me, I would take every reasonable step to ensure that the apparatus is as safe as reasonably possible. Having said all this…if the apparatus is only intended for personal use in his own backyard - and not for use by gymnasts in a workplace environment - then the legal onus of ‘fitness for intended purpose’ is relaxed.

Legal disclaimer: Anything that is homebrew or home-made by definition may come without an inspection certificate or gymnastic association approval. Most western nations have a legal concept for products known as; ‘Fit for its intended purpose’.
This takes on a particularly acute definition when children are involved. In many nations, a child (or ‘minor’) is a person who is under 18 years of age. Some States in the USA - eg Colorado + Mississippi, I think it is 21).

The comment re noise is also not relevant if only the upper connective section is metal - again, sewn webbing with adjuster buckles would work well for the lower section (feet).

Mark G
EDIT…Legal commentary added

Did you see the youtube link the OP gave?:

https://www.youtube.com/watch?v=98sr7TO7t94

The wild breakdance moves could easily rattle chain above as things recoil and make noise.

Just standing up, one’s head may be near the upper section, and so the little crevices where links meet could be an annoying long hair grabber as you reach up to connect or disconnect the system.

Since the segments involved are so short on the upper section, I don’t expect stretch to be a major issue.

Did you see the youtube link the OP gave?:

Clearly Roo, you have a little difficulty understanding my words. I thought I was clear in that I stated that I was not 100% certain of how and the manner in which the apparatus would be used. I dont have access to youtube on my computer due to site blocking but I will try to use someone else’s computer if I find time.

You seem fixated on pushing the hair entrapment issue…I wonder what the real source of your posts are (hmmmmmm). Maybe your various posts have some link to me not recommending your gnat hitches?
If you are intent and motivated on pushing this further, I would request that you show me the evidence you have gathered where the alleged gymnastic movements could get (hair) caught up in wire and chain above the horizontal bar. Have you personally experimented to model the hair entrapment issue? How do you reconcile a gymnast having long hair that is untied or unsecured - would that make sense when using the apparatus? I’d like to learn how you imagine a person would want to allow unrestrained/unsecured hair to deliberately flail about while using the apparatus.

As stated, the lower leg support section ought to be sewn webbing - preferably adjustable to allow the gymnast to achieve the perfect fit.

As for stretch, the greater the included angle, the greater the force - its simple physics. Wire or chain would work. And yes, to ease your mind - webbing straps with sewn terminations would also work.

I’ll make my position crystal clear: I don’t recommend the use of unsecured hitches in any apparatus that is used in a workplace or open public domain. Sewn terminations on webbing straps would be preferable to unsecured gnat hitches.

This is my last direct reply to Roo - as it is trending off-topic and because he appears (in my view) to have motivations that are not solely directed at assisting the OP.

Thank you for letting me know. That is why I asked.

You seem fixated on pushing the hair entrapment issue...I wonder what the real source of your posts are (hmmmmmm).
??? I'm just following the flow of the thread. People ask questions, respond, clarify, etc. Try not to be so suspicious.
I'd like to learn how you imagine a person would want to allow unrestrained/unsecured hair to deliberately flail about while using the apparatus.
If you look again at my previous post, I mentioned a person's head being near that portion when you go to remove or install the device before or after use (not during).

I’ve had to help people with long hair get untangled from chain on more than one occasion. It doesn’t happen regularly, but when it occurs, it’s a mild annoyance.