Why mathematicians study knots(quantamagazine.org)
quantamagazine.org
Why mathematicians study knots
https://www.quantamagazine.org/why-mathematicians-study-knots-20221031/
18 comments
The article seems to be implying that mathematicians study knots because there's lots of fun to be had. This is partly true, but there's other more profound reasons. For example: higher dimensional algebra. The "usual" algebra is just one dimensional, but people have noticed that in many cases this is just a projection (shadow) of higher dimensional systems, where the symbols can interact in more than just a linear direction. And once you get to three dimensions hey presto your algebra can get knotted!
Also, knot complements are an important class of 3-manifolds.
just a little further than that and every 3 manifold is surgery on a link.
I would love to have some pointers to the higher dimensional algebra you're referring to.
John Baez has written a lot about this stuff. Try this: https://arxiv.org/abs/0903.0340 . There's also his TWF column.
https://youtu.be/EBWP1POPc2A and the associated knot theory from that university - https://www.youtube.com/c/MathatAndrews/search?query=knot
One of the problems of old on CodeGolf.SE - Knot or Not - https://codegolf.stackexchange.com/questions/30292/knot-or-n...
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.
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.
Did you try a representation that's a graph where the nodes are crossings (colored for over or under), and there's an edge between any two crossings iff there's a string that directly connects them (not going through any other crossing first)?
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.
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.
It's been too long and I'd have to look at it again.
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.
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.
Yeah that seems pretty tricky. I'm somewhat convinced that the graph representation will work (it can represent the input correctly), but I'm not yet sure if it's actually useful or not for an algorithm to work through.
It needs to be in a format that you can then use existing algorithms with - the Reidemeister move being the "easiest" https://en.wikipedia.org/wiki/Reidemeister_move
Yeah that's along the lines of the transformations I was thinking of, though I only considered 2 of those...
I worked on a fun project involving knots once:
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.
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.
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.
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.
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.
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.
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.
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.
This is a bit head-spinning:
> "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:
https://www.youtube.com/watch?v=h68AxRiGxBw&t=0s
> "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:
https://www.youtube.com/watch?v=h68AxRiGxBw&t=0s
Because they are nerds. Duh.
I see this comment getting downvoted, but I agree with the sentiment; it suffices to say that if there a compelling mathematical structure exists, there will be people studying it. I imagine most graph theorists are hardly interested in the applications of their field, they just happen to be studying a structure which has much more intuitive practical applications (social networks, molecules, etc).
> there will be people studying it.
"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]
"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]
As a mathematician with a passing interest (read: no results or publications) in knot theory, yeah, we're nerds and proud.
No, "people studying abstract mathematical structures" == "nerds." I say this as a proud nerd.
Having known a ton of mathematicians, I'm not at all convinced this holds (although it undoubtedly is a subset)
Yes. Nerds are people who show more than normal interest in non-mainstream subjects like Beyonces haircut.
Kinda sorta related. I have a copy of ABOK on the coffee table and a half dozen or so 6' lengths of 12 str and polyester climbing line on the shelf.
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.
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.
What, you knock your guests out with the book and finish them off with the cord?
Interesting topic, but I don't find the answer to the headline in the text.
You're sort of right.
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?
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?
> protein folding overall - is knot theory not used there?
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.
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.
The answer is in the concluding paragraph.
>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.
>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.
As for knots - they say there is a significant interception in the center of the venn diagram of shibarists, scouts and climbers, not that sure about mathematicians though
I'm amused at the notion of a mathematical "shibarist" (I don't know Japanese but I understand that -ka might be the proper suffix). You've got some kinkster who wants to get tied up, but the knot nerd keeps trying to compute invariants...
I've found that knowing half a dozen common knots has helped me a lot more in day-to-day life than any of the mathematics I ever learned.
always relish a chance to share one of my favorite papers ever
https://www2.math.upenn.edu/~ghrist/preprints/silnikov.pdf
https://www2.math.upenn.edu/~ghrist/preprints/silnikov.pdf
How kooky is this theory? It's a proposed unified theory based on knotted spacetime.
https://knotphysics.net
https://knotphysics.net
IIRC another application is Telescent's nifty programmable/robotic physical patch panels, which apply knot theory to avoid tangling the fiber optic cables.
Dunno why we can't just accept that some mathematicians are knotty without trying to ascribe meaning to it.
IN hackin roblox