Seeing knots as curiosities of topology seems to miss the entire point of knots: to fasten something with varying attributes of tightness, slippage, tension, time, application constraints (E.g. throwing lines from a distance over cleats/bollards) (etc....). I have spent some hobby-level time playing with knots and it's curious to see inherited wisdoms from the sailing world and consider the engineering evolution of a knot being ~perfect for its precise application. It's quite a marvel to see what can only be described as the minimally pure and evolved knots, like the bowline. There are many attributes at play and fun constraints to consider. Seeing knots as fun puzzles in only the topological space doesn't seem to account whatsoever for their tensile characteristics ... which... uhh is the entire reason we use them.
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You can think of the "topological" knot as an abstraction of the physical knot in the sense that you can take away all the physical properties of the knot and still be left with certain structure that captures the "essence" of the knot. This is what mathematics pretty much is. If you want to keep other properties of the rope, that's all cool, but then you start doing physics more than mathematics.
Ah thanks that makes it a bit clearer. I guess I (falsely) see mathematics as the “purer” manifestation of a thing and thus hold it to account to abstract the complete substance of a thing and its properties. But as you say, the physical is of physics and that is where I should look for a more exhaustive abstraction perhaps.
That's like saying mathematicians miss the entire point of numbers, which is to enumerate your head of livestock, or bushels of corn.
You got it backwards. His use of knots is a bit more complex than our current mathematics of knots can explain, if we talk about knots in the 3D space.
In fact, we don't have a predictive model of knots that can in the general case say if one knot has better tensile properties than the other knot.
Is this a critique of knot theory? Far from it. It is an open question, an invitation to do new theories, which can only be answered with exploration.
If we talk about the engineering properties of real knots in 3D space, then that is something entirely different from the topology of ideal knots, which has to do with enumerating the unique ways in which they loop around an intersect themselves. Those properties don't change even if the knot is left entirely loose, so that the string never touches itself.
Is it true that current mathematics cannot explain what's going on in a 3D knot in space? Even if a closed equation couldn't solve it symbolically, surely we must have have clever people who can build a finite element model of it and do it numerically.
Very true. FWIW I’m not an academic by nature so wretchedly practical perhaps and probably impatient with what I see as narrowly applicable studies.
I got a knots app to learn a few basics that I need. One of my needs is tying up a small motorboat to a dock. I used to improvise, and just go crazy cross-crossing the cleat with the line many times, and then wrapping the line around the cleat, and on and on.
Then I learned about the cleat hitch, and watched a video to see why it worked. So simple, and it does the job perfectly. It seems too simple and light to work, but it does. And the bowline. Another very simple, elegant solution to a common problem.