My real interest which I haven't seen much literature about is generating real-world knots that have good properties. For example if you look at the various knots, some knots have nice properties like "easy to untie" and "does not get tighter under load", which has huge impacts. These properties derive from the topology but also the physics of the knot. Would be nice to find a new hitch knot that worked better.
Yes, like the PTGL - https://bio.tools/ptgl - or, er, TOPS diagrams. The main relationship to knot diagrams is really the chirality of beta-alpha-beta motifs (the majority of which are right handed).
(I have no idea how my brain can remember a paper from 20+ years ago but not enough to find it in the literature)
Math knots embed circles while real knots are typically made with free ended ropes (although some knots are not). Math knots ignore friction and the width of the rope; real knots can't ignore friction and the width (and other phyhsical properties) matter. See the comment in this article https://en.wikipedia.org/wiki/Overhand_knot
While researching my answer I found out there's a term for what I wanted to do, https://en.wikipedia.org/wiki/Physical_knot_theory
I've had this discussion with mathematicians a few times now and they don't see the difference, so maybe I'm just missing something important.
Some thoughts on application of this knowledge would be to look at the patterns as you have described as cross-sections of rope weaving with the circular looming as the individual bobbins/spinny-things in an industrial loom - so rather that just woven-sheeth and full spin style cabling, one might achieve some really incredible properties in the woven elements along a axis such as these represent.
Especially if you further differentiate btwn material and woven state (Are you weaving in an already spun set of filiments? What are the materials for the various inputs, and even further - imagine you have a set of elements in the loom where youre certain threads are the static, more rigid scaffold - like woven titanium strands which then feed into another loom which is weaving in the kevlar or other materials including a core of optics which is protected by the outer woven sheath from these patterns of 2D knots stretched out along an axis - certain elements can be printed such like the articulating spine of a snake.
It could make a machinable-high-tensile strength cable with an optical core with protected turn radii (titanium snake spine)
See here for reference to advanced cabling:
https://en.wikipedia.org/wiki/True_lover%27s_knot
(It isn't the smallest non-alternating knot. This page says 8_19 is the smallest. https://mathworld.wolfram.com/NonalternatingKnot.html)
A more troubling example of imposing this world view is decomposing 3-manifolds, where people like to think of S^1 x S^2 as prime. This only makes sense as a form of respect for one's elders.
One constructs 3-manifolds by gluing irreducible manifolds together after cutting out balls; one needs S^3 itself to express self-loops in this gluing graph, if there are no other pieces. S^1 x S^2 is S^3 with one self-loop. But the interest is in the pieces, not the graph (which can be freely reorganized), unless one aspires to be a number theorist.
It’s so satisfying, I wish I knew an easy way to explain why.
