Why mathematicians study knots
quantamagazine.org
quantamagazine.org
I took a crack at it one time (even ordered a back issue of Scientific American - https://www.scientificamerican.com/article/the-theory-of-kno... ) and didn't even get something that had a working internal representation that I could use. It's a hard problem.
That's what comes to mind to me at least. It seems like it should at least preserve the important parts of the input and be possibly useful to work on. I haven't figured out the algorithm yet though, so maybe that's misguided.
Consider the question of "what internal structure would you use to represent the trefoil knot?" - https://en.wikipedia.org/wiki/Trefoil_knot // https://mathworld.wolfram.com/TrefoilKnot.html
And then figure out how to convert an ascii art image that represents that knot into that internal format... and you do a "I'll do it later" which never comes around.
https://arxiv.org/pdf/1004.5127.pdf
This uses knots as a mathematical structure in a quantum cryptosystem. I tried to make the paper accessible without letting it get bogged down in irrelevant details.
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.
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.
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.
> "However, in four-dimensional space we can knot spheres. To get a sense of what this means, imagine slicing an ordinary sphere at regular intervals. Doing so yields circles, like lines of latitude. However, if we had an extra dimension, we could knot the sphere so the slices, now three-dimensional rather than two, could be knots."
After a little searching, here's an animated video on how to construct a 3D representation of a 4D knot... very cool and strange stuff:
"people studying something" == "nerds" ?
[edit: i was probably too offhand about that, but what I meant is that the fact that people are engaged in something like this misses at least the conventional usage of "nerd", if wasn't obvious]
Not only is it practical to have a handful of knots and hitches memorized, it's a great way to kill some time. When people come over for dinner or drinks, the book and the cord often come out.
In one sense, the answer is right there in the subtitle:
> knot theory has driven many findings in math and beyond.
The article then proceeds with mentions of possible applications in Chemistry, "to understand the makeup of matter" (and makes no mention of protein folding overall - is knot theory not used there?).
But other than that, the paper is just a gentle introduction to knots, with little to no direct relation to the title.
Perhaps an editor thought a "Why ..." title was better/more clickbaity?
To my knowledge, no. There has been some analysis done on knots in folded proteins:
https://www.umass.edu/microbio/chime/knots/index.htm
(particularly by Prof. Taylor) but this is not the same as using knot theory for analysing the folding process or for predicting folds.
>It began as an applied area of mathematics, with Thomson attempting to use knots to understand the makeup of matter. As that idea faded, it became an area of pure mathematics, a branch of the intriguing and still unpractical domain of topology.
Basically because they're interesting. A better headline will've been something like "knot landscape in mathematics" since it covers the knot theory history and advancements.