Wild knots
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- Опубліковано 26 вер 2024
- A video about some infinitely complicated, fractal knots.
The paper by Ralph Fox is:
"A remarkable simple closed curve", Ann. of Math., (2) 50, 264-265.
www.jstor.org/...
Hsin-Po Wang ( / symbolone1 ) came up with this awesome animation of undoing the wild slipknot (which is, of course, impossible):
www.desmos.com...
Wild knot:
Shapeways - shpws.me/TL3p
Printables - www.printables...
Wild slipknot:
Shapeways - shpws.me/TL3q
Printables - www.printables...
Background music by Quiet Bison:
open.spotify.c...
/ quietbison
The Achilles image is from en.wikipedia.o...
As a crochet artist, I can attest that even a finite amount of slip knots can be impossible to frog (undo). Eventually the yarn frays to the point that it felts into itself and becomes a tangled nightmare
I felt that...
Pun not intended
you forgot to use infinitely thin yarn
As a fisherman, i too can attest to this, all birds nests(giant tangles because of a bad cast) are theoretically slip knots, but very rarely can you just pull and have them come undone.
I love how in the intro, the real life footage perfectly matches with the simulated footage afterwards. What I believe he did was reverse the footage of the first clip, first starting off with the 3D model in the precise position and then picking it up and turning it.
That was the trick, yes!
@@henryseg so sneaky and elegant!
0:23 "Are we allowed to tie infinitely many tangles in a piece of string?"
My wired earbuds: "Hold my topology!"
In germany we call this Kabelsalat (Cable salad)
The mention of slip knots here makes me wonder how different/similar a theoretically infinite chain of crochet would look compared to this knot
It actually looks almost exactly the same if you just do an infinitely long simple crochet chain, the only difference is that it doesn’t keep get smaller with the crochet chain since each slip knot is the same size as the last one till you run out of yarn.
And the mathematical wild slip knot connects the ends together
mccme
The blurred border between animation and camera footage makes a distinctive style. I love it, thanks for the video :)
I love hor mathematicians will name things "wild" and "tame" instead of using more intuitive terms like "unending" and "terminating." Gotta find levity where you can
Those terms are used in math elsewhere, and might be misleading in this case. Isn't every knot "unending", since it's a closed loop with no ends? Better to make up new terms without existing semantic baggage
Both "wild" and "tame" knots are loops (images of the unit circle). "Unending" could mean a circle, or it could mean an infinite line... "Terminating" suggests "has endpoints" which is not what we want...
I can see a mathematician calling something "ending" and "unterminating" just to spite you
I would not be surprised to hear that John Horton Conway came up with those names.
There are a bunch of related problems where there are “wild” and “tame” versions of a type of object, and lots of them are connected. For example, quiver representations and Lie algebras have wild and tame versions that are related, to the point where if every problem in one can be reframed as an equivalent problem in the other. I imagine this is part of that same class of problems. In quiver representation theory, you can get infinite representation “complexity” out of surprisingly small, finite quivers.
This is the first time I see this kind of intuitive explanation for the fundamental group of a knot and the Wirtinger presentation. Having it explained on a physical objects really does make the definition obvious. Apart from that, the video feels like the a short honest canonical bridge between current popular knot theory and knot research. I might just send this to a few friends of mine!
00:11 that was the most seamless transition to animation ever seen.
I know it isn't the thing that makes the infinite slipknot interesting, but it is just a simple crochet chain. if you had it loop back on itself, then it would form a true knot that looks like a long thin band.
The cool thing is, this crochet band could be joined in a way that the band itself is knotted! There is potential for some fractal knots here.
You can't loop it back in on itself. The thing has only one actual end.
This is called a "satellite knot".
@@NoLongerBreathedIn I think I'd need a picture to believe that you get a satellite knot this way...
@@saulschleimer2036 I interpreted it as "take a chain of sc and have it loop back on itself so that it is endless" - so a tame knot - "and then knot it". That definitely produces a satellite knot.
If you make a looping crochet knot, it would be a tame knot. Though you could make a wild crochet knot out of another wild crochet knot.
Given the second "obviously" trivial knot which turns out to be non trivial, the question becomes: is there an "infinite regular oviously non trivial" knot which turns out to be trivial?
Mathematicians discover crochet
is it truly tri-colorable?
It certainly is *locally* tricolorable for any *finite* section of the knot. But once you "get to the end" as it were (which, of course, is never), who is to say the color of the long strand matches up with the long piece that's basically wiggle-free?
Either way, very cool knot!
If it were not tricolorable you would be able to point to a crossing where the coloring in the video fails.
@@matthewbolan8154 Wouldn't it need to be completely coloured before you could attempt to do that?
@@Barnaclebeard If it were not completely colored you could tell me a strand in the knot which is not colored by the scheme
That is a fascinating and subtle point, as you run along a strand in an ordinary knot, its taken for granted that you can refer to the "next crossing", but if there is infinitely many crossings this may not be true! There can be crossings ahead, but for each crossing ahead, there is one that happens earlier. It feels like there is a phantom "limit crossing" at the end, but in the technical definition of tri-colorability, there are no conditions on the colors of such a thing.
That is not how mathematical induction works
7:44
Hey Henry, just wanted to point out a very pedantic detail here: namely, there are examples of nontrivial group homomorphism from a commutative group to a noncommutative one (for example, if G is commutative and N isn't, then f: G→G×N where f(g)=(g,e), does the job), but there doesn't exist a nontrivial _onto_ group homomorphism from a commutative group to a noncommutative one (which seems to be the case for this example).
Now that I've gotten that pedantry out of my system: I really love your videos 💕. Keep up the great work!
Yes, I should have said onto homomorphism!
@henryseg Thanks for sharing Henry. I don't mean to bother you but I hope you can respond to my other comment or email when you can. Thanks.very much
I very rarely get lost in math videos, but the part starting around 7:05 required an inordinate amount of work to digest.
I finally got there. But even with all the prerequisite knowledge, it was a struggle to peel through that terseness and fill in all the glossed-over details.
Guilty as charged... Defining the fundamental group properly would require a whole video of its own. For that matter, not every viewer will even know what a group is. And to really make tricolourability work in this context gets into representations of a group. So, yeah, it got technical.
OMG all the printed out knots are so adorable, i want them!!
i feel like this is more a demonstration of the limits of the definitions used and how they're applied, rather than proving that the infinite knot is not the unknot.
Always love some group theory!
That's interesting. Just looking at the wild knot, I would've been quite confident that it is related to the unknot by a homotopy through knots - is this a case where the notions of "homotopy that is a knot at each point in time" and "isotopy" actually differ, in that only the second one fails here? or am I wrong in my assumption too?
Actually, I fail to see why there can't be an ambient isotopy as well. The ambient isotopy that undoes the nth tangle, starting from the state where the first n-1 have already been undone, is the identity everywhere but in a very small region of the space; if you concatenate all of them, so the first tangle gets undone in [0,1/2], the second in [1/2,3/4] and so on, the regions on which the map is not the identity get smaller and smaller as t -> 1, so I find it hard to imagine that it isn't continuous even at t=1. I'm also not sure the proof in terms of the fundamental group convinces me - the homomorphism is just defined in terms of local crossings of the loop with the knot, but for a wild knot like this, it's not clear to me at all why this results in a well-defined homomorphism.
Edit: I think I get it now - the isotopy wouldn't be continuous because intuitively speaking each tangle pulls some of the air behind it with it when you undo it, including some of the air already pulled along by the previous knot, and some of it would have to get pulled all the way to the limit point during that process which it by continuity can't. So that makes it seem at least plausible that the knot is actually not the unknot - I am still sceptical of the two proofs sketches given in the video though.
@@peabrainiac6370 Henry included a reference to Ralph Fox's paper. See around 4:06. You can find more details there.
This might be one of the most amazing things I’ve seen. I’m starting research in mapping class groups but knot theory has been calling my name. Time to get a book on it.
Interestingly, there are connections between mapping classes and knots. This happens for "fibered knots" (or "fibred knots" - there are various spellings in the literature).
I’ll have to look into this, maybe I’ll ask my advisor. Thank you!
🤯 As always, excellent work, Henry.
ah. you've got to my small corner of mathematics, knots and how to represent them computationally
This is such a lovely and concise video. Thank you! :D
0:35 oh my Euler, this is the best Math toy ever ...
Very nice and smooth presentation, thank you!
I had a tough time following the undefined jargon towards the end. I’ve been curious about knot theory so this was very interesting.
Yeah, I couldn’t give more than a taste of the more technical stuff. There’s a good chunk of a course in algebraic topology in really understanding the fundamental group.
*what in tarnation*
One thing that jumped out to me is the difference between infinite processes and infinite states. 0.9... is 1 because it's not an infinite process of something writing the number 9 on end, merely approaching 1, but every single infinite 9 is already present. However, the issue with the infinite slip knot is you can't undo all of it at once. You have to undo one slip before you can undo the next. IDK if that is actually accurate, but is one of the ways I parse infinities
As someone who doesn't understand group theory at all, I love the idea that group theory links so many totally disparate concepts. Does this mean you can create a knot that retains the symmetry operations of each of the 219 space groups for 3d crystals?
this is awesome
Beautiful visualization of a group homomorphism
Shoutout to all the Furries who are fascinated by stuff like this.
what?
get your mind out of the gutter :3
My mind is constantly swinging between in the gutter and plato_and_aristotle.tiff
This is the big difference between nälbinding and knit/crochet - if you don't bind off the end of a knit/crochet piece and tie the beginning and end together, you can pull the entire thing apart. Nälbinding, however, cannot be undone without having access to at least one cut end.
Thank you so very much for making these fascinating videos!
Alright so, I was following for every step of that proof, except for the one step where I understand how it all links together (pun intended)
That is fascinating, though! I really thought the wild slipknot would just be an unknot. I do wonder what would happen to the knot if you threaded the return thread through the last loop in the chain. It would probably be nothing too interesting, but at the same time knot theory is a really interesting and unintuitive part of mathematics
Amazing video! Thanks!
Looks like a simple chain from crochet but one that gets smaller as it goes on.
that knot is very similar the chain stitch basic to crochet and other fiber arts. segerman should talk to crocheters
I am told by a crocheter that it is precisely chain stitch :)
Perhaps it's better to say that the contradiction is that there is no *surjective* homomorphism from Z to S_3. Of course it's possible for an abelian group to map to a non-abelian group, it's just that the image will be contained in an abelian subgroup
Either way I think maybe it's better to argue that there is a homomorphism from S_3 to the knot group instead. Because it is unclear (to me who is not a topologist) how the other more global relations in the knot group are compatible in S_3
All of the relations in the knot group come from the Wirtinger presentation - there aren’t any others. But you don’t want to map from S_3 to the knot group because the knot group has infinite order elements but S_3 doesn’t.
@@eddielam2875 There is no non-trivial homomorphism from Sym(3) to any knot group - knot groups are torsion free.
I regret having watched this.
A well done Reidermeister maneuver.
Why would you regret having your mind blown with mathematical facts? 🥸
@@StefanReich I am being a bit ironic. Maths should be easy, it is simple logic. Then some guys knot it all up like this.
1:29 that's pretty much what a synthesized kick drum looks like 📢
Supertasks allow infinitely many actions to be performed in a finite time while each action takes a non-zero about of time. I'm not sure how that wouldn't allow you to remove all loops. Also I'm not sure these properties work when you throw in infinity. My intuition says that these properties might not hold when infinity is involved.
The number of Reidemeister moves required is countable infinity so it is theoretically possible to remove all loops.
The ambient isotopy of the knot (its motion in three-space) does not change the fundamental group of the knots "complement". Note that the ambient isotopy _can_ perform supertasks, but the stages of the supertask need to happen on smaller and smaller scales. See the discussion in the video about the (modified) topologist's sine curve.
EDIT: Here is another answer. Look at the string held by Henry at timestamp 8:47. Pretend that it is made of rubber and can stretch. He holds on to the string and you preform the supertask - you undo the k^th "bite" of the slipknot in time interval [1/2^{k+1}, 1/2^k]. If you draw the pictures, you'll find that there are 2^k points of the rubbery string that are now distance 1/2^k (say) from the wild point. So, in the limit, there are infinitely many points of the rubbery string in contact with the wild point. But an ambient isotopy can't do that...
2:48 they are so cute
i know knothing.
love your work!
your slip knot explanations are interesting. i realize what you're trying to say, but a lot of these break down with an infinitely complex knot.
What else do we know about the fundamental group of the infinite slip knot? I'm assuming it's not finitely generated, but I could be wrong. Are other knot invariants like the various knot polynomials also well defined on this knot? Also it would be nice if you included references somewhere.
Henry did include a reference, to Ralph Fox's paper. See around 4:06.
@@saulschleimer2036 Thanks! I was looking in the description and at the end of the video.
Thanks for pointing out that it wasn't in the description - I added the details there.
Wonderful!
Infinity is very hard to think about, very easy to make "logical" errors with it
nice, need more videos like this
I'm fairly certain that the fundamental group of the knot complement here is just Z: let K be the knot, p its "limit point", i.e. the point where it fails to be tame, and γ any loop in S^3\K. Then since the image of γ is compact, there must be a minimum distance ε between γ and K. The set of all points of distance less than ε to K doesn't have to be necessarily nice itself, but I'm quite sure it contains enough wiggle room to construct a set K' in it that contains K and has a filled-in torus as its complement. Then γ is also a loop in S^3\K', and thus homotopic to a loop that just wraps around K' a number of times in the simplest possible way. K' of course depends on γ, but I think it can be chosen in such a way that the generator of π(S^3\K') is always the same loop in S^3\K - and that loop must thus also be a generator of π(S^3\K).
You introduced p but didn’t say anything about p?
Why must it be possible to pick K’ s.t. its complement is a solid torus? Feels like there might be something like 3 parts of K’ connecting near p?
Or...
Hm,
confusing.
Edit: like, if you took the epsilon/2 neighborhood of the knot, then near p, you have the uncomplicated part coming out, and also some number of tubes going out the other side,
But I’m not sure exactly how many, but at least two I think. If it were only one, I would think that would mean you could separate out all the crossings before that part, which I don’t think one can?
@@drdca8263 Right, I was originally going to say something about there being an open ball around p that does not intersect s, but then switched talking about ε and forgot to take the first sentence about p out. I also think you might be right that you actually can't construct K' like that, so maybe I have spoken too soon there. I will have to think about it a little more, so sorry for my hasty comment.
@@peabrainiac6370 If you reach a conclusion, I would like to read it, so if you do, I would appreciate it if you commented your further thoughts on the matter (in this thread) [:)]
A bit of healthy skepticism is, well, healthy. But please remember that any proof you come up with has to explain the "error" in Fox's 1949 paper "A remarkable simple closed curve".
When untying a slip, the knot has not changed mathematically. So it is also possible to do this move, that has no effect infinite times and still have the same knot.
Infinity sometimes makes stuff ambiguous.
The infinitely long Slipknot looks like two "bites" or half loops, with their ends trailing off to infinity.
havent even seen it yet, loving it already 🤗
The idea that the slipknot isn't untieable is really wordplay, knot mathematics.
If you were to take an un-knot, record yourself tying it into an infinite slipknot, and just played that video backwards it would be the solution on how to un-tie it. Sure playing the video backwards would take infinitely long, but it would take the same amount of time as playing it forwards. Why do you tolerate the fact that it takes infinitely long to tie it, but knot infinitely long to untie it?
"Imagine an infinite slipknot"
"But how could such a slipknot be made? It would require an infinite amount of actions to tie, and you cant have an infinite amount of actions in real life"
"Suspend your disbelief about making the knot, this is just a hypothetical"
"Ok"
"Well, such a knot would knot be untieable, because it requires an infinite amount of actions, which isn't possible in real life"
Like, we're suspending our disbelief of infinite amount of actions required to make the knot, but knot suspending our disbelief for the infinite amount of actions required to untie the knot?
If you selectively apply rules of logic, then you can prove anything. Which is useless. By this logic, the un-knot with infinitely many wiggles in it is also a knot?
Basically, Im saying that the "finite steps" requirement only applies to knots that could be tied in a finite number of steps.
Another way of thinking about it, is that to tie a knot you have to take an un-knot, cut it, and re-attatch it. But you dont have to cut an un-knot to make the infinite slipknot, so it's still an un-knot.
You have to consider the specific definitions being used. He isn’t saying just “you can’t do it because it would require infinitely many steps, and one can never do infinitely many steps.” . He is saying “doing it would require infinitely many steps, and in this context, with these kinds of steps, doing those infinitely many steps would not be/produce the kind of transformation that we have defined as being an un-knotting.” .
@@drdca8263 Again, its just word play. The finite steps was intended to be used for knots that took finite steps to create.
Again, an un-knot with infinitely many wiggles is not a knot, as he says in his video. But it would take infinitely many un-wiggles to get it not wiggly. So, by your word-play definition that makes it a knot?
@@NGabunchanumbers Well, by the standard definition, the unknot *is* a knot. It is just the trivial knot. A knot is a continuous embedding of S^1 into R^3 (or S^3) (or, an equivalence class of such things under either isotopy or ambient isotopy, or something along these lines.)
But, I think the question you mean is, under what conditions is a knot not the unknot (under what conditions is it “knotted”).
It is equivalent to (or is) the unknot, if and only if there exists an (isotopy or ambient isotopy or something along these lines, I don’t remember exactly).
So, it isn’t the unknot iff there does not exist a (isotopy or ambient isotopy or whatever) between them.
(I don’t recall the precise definition. I don’t study knot theory, and especially don’t study wild knots.)
Also, I should mention, I didn’t entirely follow the argument presented in the video, so I don’t quite understand why there doesn’t exist such an isotopy or ambient isotopy or whatever. I would have kind of expected there to be one.
All that being said, you are free to define an alternative notion of “the same knot”, as long as you make it sufficiently precise. Whether people will find whatever notion of “the same knot” you define, to be interesting to study, depends on what definition you come up with. But you are welcome to define it as you like, as long as you are precise.
I mean, I suppose you can also be imprecise, as its a free country and all that, but if you don’t make your definition precise, then you’re doin’ it wrong.
Sure. But does John Bonham have a bass drum? Mind blowing.🤯
I ran a few of the implications in reverse and came up with this core question: Is there a finite set of knots which are homotopic/isotopic to themselves after a *finite* sequence of Reidermeister moves? This seems like a situation where infinity is getting in the way.
Reidemeister moves (performed on a diagram) do not change the isotopy class of a knot represented.
A very interesting thing to think about, for sure. And very unintuitive in my opinion lol.
Actually, even if it were not a knot at all in that case, would you run into a similar problem trying to undo that wild knot even after cutting the end of it? You would still have to undo the entire knot, right? Is that another way you could make a knot, like a way of saying it cannot be undone completely?
I was nodding along happily until 7:06, and then it turned into topology word salad and you completely lost me
But is your 3D printed approximation that caps off the unknot?
I understood very little of the back half of this video, but the knot is very pretty
Is it possible to construct an infinite slipknot of finite length? Perhaps if each cell was half the length of string compared to the previous, for example. Then isn't there an analog to Zeno's paradox wherein pulling on the string at a fixed rate for a finite amount of time will undo each successive cell in half the time of the previous, untying the knot in finite time?
Yes, it is possible to construct an infinite slipknot of finite length. (The example in the video has this property, but that is not emphasised here.) No, it is not possible to perform a "ambient" isotopy, even by undoing the next cell in half the time of the previous. This is because the fundamental group of the wild knot's complement is not the same as the fundamental group of the unknot's complement.
EDIT: Here is another answer. Look at the string held by Henry at timestamp 8:47. Pretend that it is made of rubber and can stretch. He holds on to the string and you preform the supertask - you undo the k^th "bite" of the slipknot in time interval [1/2^{k+1}, 1/2^k]. If you draw the pictures, you'll find that there are 2^k points of the rubbery string that are now distance 1/2^k (say) from the wild point. So, in the limit, there are infinitely many points of the rubbery string in contact with the wild point. But an ambient isotopy can't do that...
So cool
You need infinity tangles to unknot the wild knot??! I wish I had that much time...
does that mean that crocheted items are wild knots? have i been working with wild knots?
4:20
This is an infinite crochet chain.
Greek God punishments if they found this knot:
"I SCENTENCE YOU AN ETERNITY UNTIEING THE INFANITE KNOT!! ⚡️⚡️"
I understood everything until he said "This knot is wild"
Brilliant!
There's something I dokn't quite understand. Is the Wild Slipnot knot isomorphic to the Unnot, in a topologic way? Like, I get why we canknot transform it into the Unnot in finitely many steps, but this feels to me like seeing an infinitely complex surface without holes being classified as "knot topologically a sphere" even though the only surface "canonically-without-holes" is the sphere. Does that make sense? I hope that makes some sense.
Anyway, cool video. Thank you, Henry :)
Maybe you're thinking of the Alexander horned sphere, which is a sphere, but is embedded in S^3 in such a way that one side is not a three-ball. Short answer: infinity does weird things.
infinity always fuck things up
The function at 1:29 looks like the kick drum I just made in Ableton Live.
Why does the looping pattern on the infinite knot need to get smaller and smaller?
Because it needs to converge to a point. This is similar to the discussion of the topologist's sine curve.
Hi, Henry! Where did you get those magnetic rope ends? Or did you make them? Thanks!
I made them myself years ago. I found that one neodymium paired with one regular magnet together had the right amount of force.
Can this knot be unknotted or not? No, this knot is not an unknot so it can not be unknotted.
Do you get it or knot?
It feels very uncomfortable for the infinite knot to have one property, while every pre-infinite version of the knot (with finite-N repetitions) has the opposing property.
Even more unsatisfying would be to conformal-map the knot so that the other end is now the big end... and that end would also seem to exhibit the property of the finite knot... so what, the property change occurs in the middle somewhere?? given the knot is (presumably?) constructed inductively one loop at a time, that seems wrong too!
very frustrating to think about with my finite visual mind :)
it would be interesting to see this knot being unknotted by the keenan crane repulsive shape stuff
What is the material used in the knot at 0:36?
I like the squiggly line.
So your definition of "tame" is "isotopic to a piecewise-linear knot"? I guess I'd only ever thought about tameness of embeddings of a 2-sphere into R^3 (e.g. the "Alexander horned sphere" is not tame), where the condition I'd heard was that the complement should be homeomorphic to R^3 minus S^2. I suppose I don't want to ask that sort of thing of a knot complement.
Right, Fox defines "tame" as equivalent to a (finite) polygon.
Dear Allen - in low dimensions, "tame" can be replaced by "locally flat" and both can be replaced by the phrase "we work in the PL category throughout". Note that we _don't_ want to define "tame" as "the complement is nice". Instead, tameness should be a local property. Then that, plus compactness, will imply that the complement is nice.
Note to self: don't play with your gum into infinity. That would be too much to scrape if it gets stuck on the floor :(
Eloy casagrande is in slipknot. How do you make a pi symbol?
infinite crochet link?
Thumbnail looks like the knots in Noroi
1:28 i guess it’s “knot” a big deal
You could just twist a loop infinitely many times and then that's "not the unknot" just because it takes infinitely many moves to untwist it.
It is very subtle - what counts as a move? The unknot we made from the topologist's sine wave looks like it could require infinitely many "moves" to straighten out if you choose the wrong move.
i liked this vid and its niche but versatile concept. the music was a little distracting for me. I can tell its intended implemented was to be non-distracting, so i thought this feedback might be helpful. not subscribed but looking forward to the next niche concept!
Just shake the slip knot to untie it
The thumbnail looks like a weird daisy chain.
If you treat it like a supertask, where you can complete an infinite series of actions in a finite time, why wouldn't you be able to unknot this one?
Because the ambient isotopy of the knot (its motion in three-space) does not change the fundamental group of the knots "complement". Note that the ambient isotopy _can_ perform supertasks as long as they happen on smaller and smaller scales. See the discussion in the video about the (modified) topologist's sine curve.
EDIT: Here is another answer. Look at the string held by Henry at timestamp 8:47. Pretend that it is made of rubber and can stretch. He holds on to the string and you preform the supertask - you undo the k^th "bite" of the slipknot in time interval [1/2^{k+1}, 1/2^k]. If you draw the pictures, you'll find that there are 2^k points of the rubbery string that are now distance 1/2^k (say) from the wild point. So, in the limit, there are infinitely many points of the rubbery string in contact with the wild point. But an ambient isotopy can't do that...
@@saulschleimer2036 I think I need to study this more. I have to admit here that I don't know enough about this yet to actually understand your answer.
I was seeing it as the same as counting an infinite number of balls, just having an infinite number of loops to undo.
Thank you for illuminating where my lack of knowledge is so I can go fill in that area of knowledge.
Edit, correcting auto-correct.
I agree with the other comments that this is highly unintuitive. Is there really no sense in which the unknot is isomorphic to this wild infinite slip knot? That's frustrating. Is it isomorphic to any simple knot?
If you allow isotopy of the curve in space rather than ambient isotopy (that is, continuously deforming the whole space to move the knot, rather than moving the knot on its own) then I think that the wild slipknot becomes the same as the unknot. But unfortunately, under that definition, all knots become the same as the unknot!
The infinite slip knot is not (ambient) isotopic to any tame knot. (I think that the fundamental group of the complement of the infinite slip knot is not finitely generated, but I can't find a reference. Sorry!)
Couldn't the infinite reidmeister moves be completed in a finite amount of time using super tasks? Is it because the deformation starts breaking continuity at the end?
Yes - the deformation "stretches" space more and more and "in the limit" is not continuous.
EDIT: Here is a more detailed answer. Look at the string held by Henry at timestamp 8:47. Pretend that it is made of rubber and can stretch. He holds on to the string and you preform the supertask - you undo the k^th "bite" of the slipknot in time interval [1/2^{k+1}, 1/2^k]. If you draw the pictures, you'll find that there are 2^k points of the rubbery string that are now distance 1/2^k (say) from the wild point. So, in the limit, there are infinitely many points of the rubbery string in contact with the wild point. But an ambient isotopy can't do that...
Can you have a finite complexity wild knot?
What do you mean by “finite complexity”?
It should be possible to specify a Turing machine where if you give as input a desired precision for the output, and a distance along the circle, outputs coordinates for the point that distance along some version of the knot (for some choice of base point and forwards direction)
so, if you wanted to assign a Kolmogorov-like complexity to knots, it would have a finite one.
But probably you don’t mean Kolmogorov complexity.
@@drdca8263 "Finite complexity" in the sense that this video is about "Infinite Complexity"
Literally "What if concept opposite of video main subject?"
@@youtubeuniversity3638 In that case, I believe the knot having infinite complexity is the same concept as the knot being wild.
Not just, the two being logically equivalent,
But specifically the same idea. Like, a knot being “wild” is, *I think* , defined to capture the notion of “infinitely-complicated”?
@@drdca8263 Noted, thank you.
Hey, i want to print other knots. Where can i find the files?
The files are on printables.com, links in the description. Unless you mean the other, tame, knots in the video? Those are at www.printables.com/model/167504-prime-knots-up-to-7-crossings
I like your videos, but please turn the volume off the foreground music a little down, at least until it becomes a background music. :-)
I don't agree, and part of this is pure intuitive disbelief, but fundamentally, I think the way you're approaching this problem has contradictions. Either the wild knot in question is not a knot from the technical definition, or it is an unknot, and I think both are true depending on how you define the knot and the untangling mechanism, and yes, this is one of those times where infinity causes some strange behavior.
To tackle the claim that this isn't a knot, I want you to imagine how the knot is connected on the far end of the infinite loops. You're, of course, imagining a part where the loops stop and the straight lines begin. The problem is, that doesn't exist on this knot. The loops never end, so that connection never happens, which means the knot isn't a closed loop, which means it's not a knot, and thus, your analysis of whether it's an unknot is not accurate.
You can modify the definition of this knot such that you include the entire unlooping section of the knot and both ends of the part where it begins looping. In this case, you can see both the beginning and the end of the looping section, but then you have two unconnected looping sections, because if you follow either side down, even infinitely so, you'll never get to the other end, meaning they're not connected.
However, if we assume that the entire knot is continuous and fully looped, and we know every single one of those infinite loops can be undone, then the knot MUST be the unknot. You can even define this wild knot according to how you would create it from the unknot. You create a loop, then create another, feed the second through the eye of the previous loop, and repeat the process infinitely. If you can define it that way, then the reverse must also be true.
Imagine the unknot, and then you take a section of it, create a loop, twist the loop and then feed that whole section through the eye of that loop. Then imagine that the resulting section is repeated infinite times. Of course, all of those sections are known to be unfurlable, and thus we know this is the unknot. The only difference between that hypothetical knot and the one discussed in the video is that the one in the video has each section connected to another in sequence, but we can take our existing hypothetical knot here and make it a sequentially repeating knot by just feeding each section through the next loop instead of the current loop.
The reason why the math is coming out such that it is tricolorable is simply because the knot is not continuous. A line segment not connected on both ends will always be tricolorable, but if you define it in such a way that it is connected, then it must logically be the unknot.
As pingu used to say.
"Not Knot!!"
Who's there?
This video feels like is a movie long and is boring but very interesting
math is diabolical lmao
Wild knots uwu
Is the 4D 'ghost' found in CERN's particle accelerator some kind of node? An answer please.
What? What are you talking about?
@@drdca8263 Just look for the news: 4D 'ghost' found in CERN particle accelerator
@@drdca8263 Just look for the news: 4D 'ghost' found in CERN particle accelerator
@@cortexcarvalho9423That was unwanted resonance caused by noise within the accelerator. The “4D” aspect simply refers to the use of quaternions to factor in time to avoid/lessen this noise.
I got a knot in brain now...