[HN Gopher] Why mathematicians study knots
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Why mathematicians study knots
Author : theafh
Score : 106 points
Date : 2022-10-31 16:37 UTC (6 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| shagie wrote:
| 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.
| padolsey wrote:
| 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.
| kazinator wrote:
| That's like saying mathematicians miss the entire point of
| numbers, which is to enumerate your head of livestock, or
| bushels of corn.
| padolsey wrote:
| 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.
| karmakurtisaani wrote:
| 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.
| padolsey wrote:
| 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.
| amluto wrote:
| 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.
| walterlb wrote:
| I 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.
| SoftTalker wrote:
| 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.
| andbberger wrote:
| always relish a chance to share one of my favorite papers ever
|
| https://www2.math.upenn.edu/~ghrist/preprints/silnikov.pdf
| mattpallissard wrote:
| 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.
| secondcoming wrote:
| What, you knock your guests out with the book and finish them
| off with the cord?
| fnordpiglet wrote:
| Because they are nerds. Duh.
| captainclam wrote:
| 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).
| ska wrote:
| > 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 wrote:
| No, "people studying abstract mathematical structures" ==
| "nerds." I say this as a proud nerd.
| ska wrote:
| Having known a ton of mathematicians, I'm not at all
| convinced this holds (although it undoubtedly is a
| subset)
| klyrs wrote:
| As a mathematician with a passing interest (read: no
| results or publications) in knot theory, yeah, we're nerds
| and proud.
| fnordpiglet wrote:
| Yes. Nerds are people who show more than normal interest in
| non-mainstream subjects like Beyonces haircut.
| mungoman2 wrote:
| Interesting topic, but I don't find the answer to the headline in
| the text.
| MonkeyClub wrote:
| 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?
| gilleain wrote:
| > 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.
| forgotpwd16 wrote:
| 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.
| PIENIKAKKA123 wrote:
| IN hackin roblox
| mathgenius wrote:
| 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!
| macrolocal wrote:
| Also, knot complements are an important class of 3-manifolds.
| taliesinb wrote:
| I would love to have some pointers to the higher dimensional
| algebra you're referring to.
| shagie wrote:
| https://youtu.be/EBWP1POPc2A and the associated knot theory
| from that university -
| https://www.youtube.com/c/MathatAndrews/search?query=knot
| izhak wrote:
| 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
| klyrs wrote:
| 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...
| xeonmc wrote:
| Artists do too.
| jagged-chisel wrote:
| But not for the same reasons
| ska wrote:
| Are you so sure? They are similar reasons, at least in part.
| ok_dad wrote:
| Mathematicians are half-artist, I think. Or, vice-versa.
| jansan wrote:
| Sailors and climbers too.
| adolph wrote:
| And Dons, really kept 3's Company together and Mayberry
| loosened up
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