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Why mathematicians study knots
- mungoman2 4y agoInteresting topic, but I don't find the answer to the headline in the text.
- forgotpwd16 4y agoThe 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.
- MonkeyClub 4y agoYou'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?
- gilleain 4y ago> 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 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.
- xeonmc 4y agoArtists do too.
- jansan 4y agoSailors and climbers too.
- adolph 4y agoAnd Dons, really kept 3’s Company together and Mayberry loosened up
- jagged-chisel 4y agoBut not for the same reasons
- fnordpiglet 4y agoBecause they are nerds. Duh.
- captainclam 4y agoI 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).
- ska 4y ago> 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]
- actually_a_dog 4y agoNo, "people studying abstract mathematical structures" == "nerds." I say this as a proud nerd.
- ska 4y agoHaving known a ton of mathematicians, I'm not at all convinced this holds (although it undoubtedly is a subset)
- klyrs 4y agoAs a mathematician with a passing interest (read: no results or publications) in knot theory, yeah, we're nerds and proud.
- fnordpiglet 4y agoYes. Nerds are people who show more than normal interest in non-mainstream subjects like Beyonces haircut.
- izhak 4y agoAs 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
- klyrs 4y agoI'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...
- shagie 4y agoOne of the problems of old on CodeGolf.SE - Knot or Not - https://codegolf.stackexchange.com/questions/30292/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-knots/ 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.
- kadoban 4y agoDid 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.
- shagie 4y agoIt'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://en.wikipedia.org/wiki/Trefoil_knot // https://mathworld.wolfram.com/TrefoilKnot.html 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.
- SoftTalker 4y agoI'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.
- amluto 4y agoI worked on a fun project involving knots once: https://arxiv.org/pdf/1004.5127.pdf 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.
- mattpallissard 4y agoKinda 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.
- secondcoming 4y agoWhat, you knock your guests out with the book and finish them off with the cord?
- mathgenius 4y agoThe 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!
- macrolocal 4y agoAlso, knot complements are an important class of 3-manifolds.
- ouid 4y agojust a little further than that and every 3 manifold is surgery on a link.
- taliesinb 4y agoI would love to have some pointers to the higher dimensional algebra you're referring to.
- shagie 4y agohttps://youtu.be/EBWP1POPc2A https://youtu.be/EBWP1POPc2A and the associated knot theory from that university - https://www.youtube.com/c/MathatAndrews/search?query=knot https://www.youtube.com/c/MathatAndrews/search?query=knot
- mathgenius 4y agoJohn Baez has written a lot about this stuff. Try this: https://arxiv.org/abs/0903.0340 https://arxiv.org/abs/0903.0340 . There's also his TWF column.
- PIENIKAKKA123 4y agoIN hackin roblox
- padolsey 4y agoSeeing 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.
- kazinator 4y agoThat's like saying mathematicians miss the entire point of numbers, which is to enumerate your head of livestock, or bushels of corn.
- padolsey 4y agoVery true. FWIW I’m not an academic by nature so wretchedly practical perhaps and probably impatient with what I see as narrowly applicable studies.
- Shorel 4y agoYou 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.
- kazinator 4y agoIf 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.
- andbberger 4y agoalways 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
- walterlb 4y agoI saw "The Ashley Book of Knots" mentioned on another post, and it has some interesting history of knots. Especially that the introduction of books led to sailors spending more time not knotting.
- abnry 4y agoThe book is also open domain, so anyone interested should look it up.
- photochemsyn 4y agoThis 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 https://www.youtube.com/watch?v=h68AxRiGxBw&t=0s
- hansvm 4y agoDunno why we can't just accept that some mathematicians are knotty without trying to ascribe meaning to it.
- scotty79 4y agoMissed opportunity to title the article "Why knot maths?".
- DesiLurker 4y agoHow could they knot?
- musicale 4y agoIIRC another application is Telescent's nifty programmable/robotic physical patch panels, which apply knot theory to avoid tangling the fiber optic cables.
- hoseja 4y agoHow kooky is this theory? It's a proposed unified theory based on knotted spacetime. https://knotphysics.net https://knotphysics.net