George Hart made a video to show you how it’s done using math, right after proving that it can’t be done, also using math. 'Tis very cool.
Some notes:
a mathematical/topological trefoil knot is an overhand knot or simple knot
if you want to duplicate what he did in the video, he’s obviously not using a rubber band. It’s called a “jelly bracelet” and you’ll find it where you find kids’/youthful fashion. Like Claires or Toys ‘r’ Us or the like. Dollar stores or Daiso would also be good candidates.
Very nice. He didn’t tie the knot. He simply cut the band in a clever way and let the trefoil that was already there emerge.
Using this method, you could make any cylindrical/non-flat turk’s head into a simple loop by dipping it in glue. And you could change it back by separating the strands again. Teoretically, you could make any turk’s head from a single band. I have whittled chains from pieces of wood, but it never occurred to me to try something as complex as a turk’s head. Maybe someday…
Well said ! However, “emergence” is creation, birth of something that comes out of nothing, the arising of a complex whole that is irreducible to its simple parts. It is not just the potential turned into the actual - does any knot emerges from any rope ?
In applying the theory of Jones polynomials, you have to realize that the string that makes the knot has zero width. There is nothing to cut. Once you take a knife to the string (rubber band), the mathematics no longer applies. If in doubt, please refer your questions to Prof Vaughan Jones FRS, IGKT Vice President, who is also the mathematician who discovered the Jones Polynomials.
Mathematics is everywhere, I am afraid… Here we are dealing with a torus knot (1) ( the (2,3), for the right handed trefoil knot, or the (2, -3), for the left handed trefoil knot).
" In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Each torus knot is specified by a pair of coprime integers p and q. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot."
“The Jones polymonial of a (right handed) torus knot is given by : ( … Oh, my knotGod ! )”.
I’m sorry, but cutting material is blatant cheating. It’s not “math”. With a large enough piece of material or a precise enough cutter, virtually any geometry can be carved out even if the procedure is more direct than Mobius-style cutting. It’s all cutting.
If cutting isn’t cheating, then it seems that Ashley did discover the Zeppelin bend. ABOK #582 looks pretty durn close to the Zeppelin bend to me. One cut (shown by the green line in the picture – blatantly copied from ABOK) and Bob’s your uncle. ;D
There is no cutting involved here… The continuous lines that run alongside each other on the surface of the torus ( and form a torus knot ) are not cut, their continuity is not interrupted - they are just detached /separated from each other.
I has become some sort of a fashion, to name anything we do not understand " a cheating ". Mathematics do not cheat, human minds do…(1)
If you “see” that the “cutting” had only to do with separating the (more than one) 1D lines that run on the surface of torus from each other, you will see that there is no surgury involved here - the unknotted line drawn on the torus becomes an knotted line in 3D.
You just do not “see” the torus, because it is not shown !
George Hart just wanted to show that the rubber band can be considered as the surface of a torus, and that, if we separate the continuous lines (“hidden”) in it, we reveal the “emergent” (? ? ? ) torus knot, which, in this case, is the trefoil.
However, 