Timeline for Solving the unknotting problem by pulling both ends of the string
Current License: CC BY-SA 4.0
42 events
| when toggle format | what | by | license | comment | |
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| Mar 28 at 3:23 | comment | added | Craig Feinstein | @RegularGraph his point is valid, but hard to understand. His picture convinced me. I commented on it in my update of the question. | |
| Mar 28 at 1:30 | comment | added | RegularGraph | @CraigFeinstein The phenomenon that AndyPutman's imgur picture illustrates is kind of interesting, but I don't see how it has much to do with your original question. (If it had been someone else commenting, I would have just ignored it, but Putman is a strong mathematician, so I'm trying to understand his point. Maybe this is a bad attitude on my part.) | |
| Mar 27 at 21:25 | comment | added | Craig Feinstein | @RegularGraph I did not edit anything out of the regular post. I only made updates. | |
| Mar 27 at 13:13 | comment | added | RegularGraph | @AndyPutman Once you cut the string, it's no longer an embedding from S^1, so speaking of the unknot seems a little strange. I'm not saying anything non-trivial, just that the procedure of testing for unknottedness by cutting the knot at one point, and then pulling on the two ends, is at least topologically reasonable. Your knot will be untied in this way, while the actual trefoil knot will not be. I would prefer the main post be a little clearer, but I think in its current state it no longer contains a topological fallacy. Do you agree? | |
| Mar 26 at 1:06 | comment | added | Andy Putman | @RegularGraph: If you cut any knot at any point it becomes the unknot, so I'm not sure what the point of that is. This knot refutes the claim on the OP that if you have an unknot, then pulling it apart from two points and letting strands slide past each other with no friction will turn it into the standard unknot. | |
| Mar 26 at 0:58 | comment | added | RegularGraph | @AndyPutman I see now, thanks for explaining. I thought you had just drawn it thickened to make it look more like an actual rope. It's still true that if you cut it at some point and then pull the ends, you can get it to untie (assuming no issue with friction etc). I came late to this discussion; I guess this was meant to refute a claim in the original post that has since been edited away. | |
| Mar 25 at 21:51 | comment | added | Andy Putman | @RegularGraph: What I drew is the whole knot. There are no ends to join — if you follow the path of it, it closes up. It has two parallel strands that seem to trace out two parallel trefoils, but then at the ends join together. So it’s an unknot. | |
| Mar 25 at 21:23 | comment | added | RegularGraph | @AndyPutman I don't understand your imgur example -- what you get when you join the ends is a trefoil knot, no? | |
| Mar 25 at 18:25 | comment | added | Craig Feinstein | @Ryanbudney the counterexample given in these comments appears to me to be impossible to undo given any material. I have tried it on magician’s rope. | |
| Mar 25 at 18:09 | comment | added | Daniel Asimov | If we use an actual curve for the knot (with zero thickness), then a merely continuous motion of pulling it tight can cause any knot to disappear. | |
| Mar 25 at 18:05 | comment | added | Daniel Asimov | A string with its ends glued together is a solid torus (not a "torus"). | |
| Mar 25 at 17:42 | comment | added | Ryan Budney | @CraigFeinstein: you still have an ill-defined question. For example, the dynamics of pulling tight will be different depending on how torsionally-rigid your rope is. For example, think of the difference between a cotton shoe-lace and a garden hose. Being torsionally stiff will increase the likelyhood of the knot forming tight self-twists, and can cause different critical points of the flow. | |
| Mar 25 at 16:58 | answer | added | Michael | timeline score: 2 | |
| Mar 25 at 13:11 | history | edited | Craig Feinstein | CC BY-SA 4.0 |
Update
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| Mar 25 at 1:54 | answer | added | Ian Agol | timeline score: 6 | |
| Mar 14 at 12:10 | comment | added | Mikhail Katz | It seems that it should follow from the work by Nabutovsky and Weinberger on logical complexity of thick ropes that such "pulling" will necessarily get stuck on some trivial knots; see web.archive.org/web/20170809004938id_/http://www.mathunion.org/… | |
| Mar 13 at 20:17 | history | edited | Craig Feinstein | CC BY-SA 4.0 |
Update 2
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| Mar 13 at 16:07 | vote | accept | Craig Feinstein | ||
| Mar 13 at 0:23 | comment | added | Craig Feinstein | @ryanbudney I’m not sure what you mean. I read your comment before. The physical attributes of the string might affect things in the real world but on a computer they can be easily adjusted. We might as well make it frictionless. | |
| Mar 13 at 0:05 | review | Close votes | |||
| Mar 22 at 3:10 | |||||
| Mar 12 at 23:42 | comment | added | Ryan Budney | I think you need to be precise in the question you are asking. Right now you have a bit of a "shotgun" question, where it's up to the reader which of many possible questions you might want answered. | |
| Mar 12 at 21:12 | comment | added | Craig Feinstein | @AndyPutman I updated my question. I still think the question is a good question even though my definition of unknot is not so rigorous. | |
| Mar 12 at 21:03 | history | edited | Craig Feinstein | CC BY-SA 4.0 |
Update
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| Mar 12 at 20:43 | comment | added | Yoav Kallus | I'm curious if in your experiments included knots (physical knots, mathematically really unknots) designed to be loaded on the free ends. The two that come to mind are the alpine butterfly and the sheepshank. A figure-8 on a bight is also worth trying, though I can see that rolling with frictionless rope. | |
| Mar 12 at 20:28 | comment | added | Andy Putman | @SamHopkins: I agree that one could ask an interesting question along these lines, but this isn’t it. I really think the whole premise (at least as described in the question) is based around a fallacy. | |
| Mar 12 at 20:20 | comment | added | Craig Feinstein | @AndyPutman now I understand. Many magic tricks with rope are built around that principle. | |
| Mar 12 at 19:40 | comment | added | Sam Hopkins | I think the downvotes are excessive. Yes, the question is vague, but there is a real, research-level question lurking here as evinced by the various references people have provided. | |
| Mar 12 at 19:33 | comment | added | Andy Putman | @CraigFeinstein: No. Here is a picture of an unknot. If you grab it at the red and blue points and pull, it will not come undone. imgur.com/a/lJKSlrj | |
| Mar 12 at 19:23 | comment | added | Craig Feinstein | @AndyPutman sorry about the name misspelling. Am I understanding you correctly that this problem may have a different answer for a rope that has two ends with a restriction on what moves can be made to unknot it than a rope in which the ends are glued together? | |
| Mar 12 at 19:10 | comment | added | Andy Putman | (you also should spell my name correctly if you want me to notice your comment) | |
| Mar 12 at 19:09 | comment | added | Andy Putman | @CraigFeinstein: Yes, but unless the string can pass through whatever is holding it it cannot become unknotted. This is why I think you are being led astray by relatively simple knots. If you tried this for something genuinely complicated it would not come undone. | |
| Mar 12 at 19:05 | answer | added | Sam Nead | timeline score: 13 | |
| Mar 12 at 18:51 | comment | added | Sam Nead | I've downvoted, but I feel I should explain. It looks like you are asking for a reference. But the reference you are asking for is "has anybody had my idea before me?" For that to be a real question, you have to place some flesh on your (currently) skeleton of an idea. | |
| Mar 12 at 17:48 | comment | added | JoshuaZ | Even with a very slippery piece of string, something which is an unknot can seem to get caught when you pull tight on it. For one which I think will show this somewhat, making this unknot out of string and pulling it tight this way after cutting the upper left i.sstatic.net/dtMM9.jpg . | |
| Mar 12 at 16:40 | comment | added | Craig Feinstein | @RyanBudney I was assuming friction would be very small or zero, since if there is too much friction it could be hard to get things untangled (from my own experience doing magic tricks with ropes). | |
| Mar 12 at 16:10 | comment | added | Ryan Budney | You've missed some key issues. You need your rope to be thick (or else you pull it tight, breaking the topology) and you need your rope to be frictionless. Is that your set-up? If so, you should edit it into your question. If you allow friction you get genuine critical points -- this is the subject of traditional knots, like used in climbing or fisheries. | |
| Mar 12 at 15:55 | comment | added | mlk | People have tried similar things, e.g.: arxiv.org/abs/2006.07859 But simulating a physical string means discretizing it and with that it is very easy to accidentally jump in topology. In addition, seeing that this process seems to work in examples is easy. But showing that it always finishes the job in polynomial time seems more or less impossible. | |
| Mar 12 at 15:50 | review | Close votes | |||
| Mar 12 at 16:06 | |||||
| Mar 12 at 15:45 | comment | added | Craig Feinstein | @AndyPutnam I don’t see the problem here. I only care about knots on the string, not my body. | |
| Mar 12 at 15:35 | comment | added | Andy Putman | What you say isn't even true. You're holding the string with two hand, so the union of the string and your arms/body can form a nontrivial knot that will not come undone when you pull it even if the actual string is not knotted. | |
| Mar 12 at 15:33 | comment | added | The Amplitwist | Related: Are there any very hard unknots? | |
| Mar 12 at 15:26 | history | asked | Craig Feinstein | CC BY-SA 4.0 |
