snip...

It would seem to me that mathematical knot theory only takes one so far (not very far?) in understanding even the structure of a practical knot. For example, the relative positioning of the elements of a knot are not considered.

DDK

True, but then 'Knot Theory' is only a tiny and highly stylised subset of the realm of Knots. Indeed, the field we study (more correctly that should be 'fumble around in') is in itself a tiny subset of the realm of knots our universe plays with. 'Knot Theory' deals only with topology, while we are attempting to get to grips with (pun intended) Force Machines within which 'Topology' is itself only one aspect.

Sadly (or not depending on your perspective), our chosen field is so mind numbingly complex, we tend only to sit around the camp fire and semi randomly twiddle cordage, then marvel at the Force Machines that knots can create.

Unlike the mathematicians, we have not yet attempted to simplify / stylise our field so that we might start to comprehend the workings of the machines we so effortlessly make, use and discard.

Unlike the Chemists, we have not yet sought to identify the component 'elements', identify their properties, understand their 'Stable energies' and the energy needed to morph them to higher or lower energy forms, to formulate how they interact with other components and most importantly, to understand how they process FORCE. A knot is a tiny analogue, 3D, real time force computer. We - make a knot - put force in - and study the final consequence - we have virtually no idea what happens within our knots, and so consequently we have virtually no idea how to rationally refine or optimise them.

But that leaves a whole lot of exploration for those minded to try to go there...

Derek