I have a theory about tying shoelaces. I tie mine in the "correct" way which doesn't reposition the loops vertically. Never thought about it until that viral TED talk. My son, however, does the opposite. Since I'm the one that taught him how to tie his shoes, my theory is that he watched me make knots, from the opposite side, and thus internalized making the same knot, but in reverse.
If my theory is true, then every other generation ties their knots in reverse...
I never learned how to do the "one bunny ear" effectively (I could never remember some detail and they'd always fall apart) and did the "two bunny ear" method until 2010 or so when the Numberphile video on the faster way came out. Still two ears, but they pass through each other instead of wrapping around.
Alas, I now wear mostly Crocs day-to-day and boot laces end up being laced more tightly in the base anyways (twice around instead of one before any loops) where they are tight even if the thicker laces get worked loose over many miles.
I was having trouble understanding clockwise vs counter-clockwise with a standard shoelace knot, so I just grabbed a pair of sneakers. Turns out I do it the "wrong way" (clockwise), but I can't even remember the last time I had a shoelace untie on me since grade school... well, except for those stiff waxy laces on dress shoes.
As an excellent application of knot theory, the video demonstrates two ways to avoid the problem of untangling messy earphone cords. The first one, which I happened to figure out by myself, is to store them confined to a tight space. The second one, which strains the cord a bit more but still appears practical, is demonstrated around 31:20.
The title is misleading click batey but very interesting video about knot theory.
From the video it looks like mathematical knots might not have the property actually of being real knots that you can tie. Is this true? Does anyone know?
In mathematical theory, the ends of the knots have to be connected. Otherwise, the knot can be untied and it's just a straight piece of string. It's not really a "knot" unless it can't be untied. A circle is the simplest knot, and is effectively the "zero" of that mathematical system.
So, the physical knots you might think of being useful in various situations have only somewhat of a relationship to mathematical knot theory.
I think there's a close correspondence though. What you naively think of as a knot is a mathematical knot if you join the ends together, and I think any given mathematical knot can give rise to a regular knot.
I know very little about it other than a friend of mine did a PhD in it, but it gets weird fast. Like they often consider it in canonical forms that look nothing like a practice knot, and then of course they start showing correspondences with polynomials &c.
> So, the physical knots you might think of being useful in various situations have only somewhat of a relationship to mathematical knot theory.
No, you can map real world knots to mathematical knots. It is explained in the video, with common examples. You can also map the other way around, but there's way more mathematical knots (prime and non-prime) than people could ever need.
To me the difference is about things like friction, diameter of the rope, and the physical reasons behind why a knot resists binding, tightening, and so on. The fact that mathematical knots don't have ends isn't that important. If you have two ends of a rope and so long as those ends never move into the knot's "territory", then its dynamic the same as a circle knot. Isn't it?
That's where it principally diverges from physics - in reality a knot is about creating friction and wrapping a rope a few times around a tree can for some practical purposes achieve the same thing.
> The title is misleading click batey but very interesting video about knot theory.
He actually acknowledges this in an earlier video, where he mentions that that the YouTube algorithm no longer gave him much of an audience with his more natural-form videos, and he explicitly said that he would try a few clickbaitey titles to feed the algorithm.
Interestingly, when I saw that I figured this channel for a goner because he was going to turn to the darkside, but quite contrary to expectations I feel like his channel has actually blossomed since then. His older videos are fine, but often just explaining concepts that you can find a lot of other explanations on, but some of his recent videos have rocked the boat even among physicists and IMHO he was ultimately proved to be correct for them. I particularly point at his videos about how nobody has measured the one-way speed of light ( https://www.youtube.com/watch?v=pTn6Ewhb27k ), which was well-known among people very very deep in the field of relativity but not very well known outside of that (even amongst physicists in other fields, let alone lay people) and the electricity videos he's done in the last year (https://www.youtube.com/watch?v=bHIhgxav9LY , https://www.youtube.com/watch?v=oI_X2cMHNe0 ) which attracted a whole whackload of "debunking" videos, which were themselves wrong (the best "bunking" video, if I may call it that, is probably AlphaPheonix's: https://www.youtube.com/watch?v=2Vrhk5OjBP8 , since he actually set up the experiment, albeit on a smaller scale).
They aren't all of that caliber, but the knot theory video was very good too and not something I've seen a hundred times elsewhere, and the airships video that preceded that one clearly had a high degree of research put into it beyond just the hype.
Mathematical knots use endlessly slippery and stretchy rope. Most real knots operate on friction one way or another.
Anyway, as a more direct answer to your question, mathematical ropes use a single loop of rope. By definition the only one that can be tied this way (without breaking the rope) is the unknot.
There are several different real knots that can be tied from an unbroken loop of rope.
That's because "real knots" depend heavily on the material they are tied in.
Many knots that work well in a hemp rope won't hold in slippery dyneema, even fewer will hold in fishing line.
Mathematics wouldn't be the right field to investigate with such physical properties, so they simplify and remove the need for those properties altogether. Mathematical rope is infinitely stretchy and slippery.
https://www.nicoleondre.com/heatwork-tanya-leighton-2023
https://www.nicoleondre.com/primes
Full disclosure, she is also my wife.