5 ms·
A simple way to measure knots has come unraveled
- NoMoreNicksLeft 1y agoMaybe I'm dumb, but they have two knots that have a number of 3, one is the mirror of the other. They were hoping that it would add up to six, but it only adds to 5. Wouldn't this mean that there is a sort of "negative" number implied here? That one knot is +2/+1 and that the other knot is +2/-1, and that their measure (the unknotting number) is only the sum of the abs()?
- drakythe 1y agoI don't think it implies that one of the knots has a negative number. It proves that when you add knots together you can't just add together their unknotting numbers and expect to get a correct answer. The article mentions "unpredictability of the crossing change" as a source of this issue (if I am reading that statement correctly). Basically the unknotting number combes from how the string crosses itself and when you add two (or more?) knots together you can't guarantee that the crossings will remain the same, which makes a kind of intuitive sense but is extremely frustrating when there isn't a solid mathematical formula that can account for that.
- taeric 1y agoI think you are reaching more for imaginary numbers? I could see trying to fit this with surreal numbers, as well. Would be fitting, as I think Conway was big into knots? Regardless, no, not dumb. Numerically modelling things is hard, it turns out. :D
- deleted 1y ago[deleted]
- scuppernong 1y agoThat would certainly be interesting, though I don't know of any matching definition in knot theory. There is a notion of "positive" and "negative" crossings, so you could define the positive and negative unknotting numbers by asking how many of each you have to swap. Unfortunately, in their example, all torus knots can be drawn with all positive or all negative crossings.
- pfortuny 1y agoNot exactly. Orientation is a bitch, just make a mobius strip anc cut it across its middle, and do it again… magic.
- kmill 1y agoAn analogy might be how if you mix together water and alcohol, you get a solution with less volume than the sum of the volumes. That doesn't mean that there's "negative" volume, just that the volume turns out to be sub-additive due to an interaction of specific characteristics of the liquids. Somehow, some connect sums of particular knots enable possibilities that let it more easily be unknotted. I spent the better part of the summer during grad school trying to prove additivity of unknotting numbers. (I'll mention that it's sort of a relief to know that the reason I failed to prove it wasn't because I wasn't trying hard enough, but that it was impossible!) One approach I looked into was to come up with some different analogues of unknotting number, ones that were conceptually related but which might or might not be additive, to at least serve as some partial progress. The general idea is represent an unknotting using a certain kind of surface, which can be more restrictive than a general unknotting, and then maybe that version of unknotting can be proved to be additive. Maybe there's some classification of individual unknotting moves where when you have multiple of them in the same knotting surface, they can cancel out in certain ways (e.g. in the classification of surfaces, you can always transform two projective planes into a torus connect summand, in the presence of a third projective plane). Connect summing mirror images of knots does have some interesting structure that other connect sums don't have — these are known as ribbon knots. It's possible that this structure is a good way to derive that the unknotting number is 5. I'm not sure that would explain any of the other examples they produced however — this is more speculation on how might someone have discovered this counterexample without a large-scale computer search.
- rep_movsd 1y agoMaybe I'm really dumb, but it should be obvious that replacing a section of rope in one knot with another, is intuitively not going to simply "add the unknotting numbers"
- AlotOfReading 1y agoAnd yet it almost always works. There were no known counterexamples where it failed until this was published.
- jlarocco 1y agoI'm with the OP on this one. Intuitively (to me, anyway) I wouldn't expect it to work in general. I'm surprised it took so long to find a counterexample, but it doesn't surprise me at all to hear it doesn't work.
- furyofantares 1y agoMine too, but this is not very interesting, although our intuition was right, it was almost certainly right for the wrong reasons.
- jlev1 1y agoAgreed, this is the real takeaway for those who think it’s unsurprising: they simply haven’t understood why it is surprising.
- jlarocco 1y agoDefinitely! My understanding of knot theory is limited to having watched a few YouTube videos and reading the first introductory chapters of a book. A neat topic, but not one I'm going to dig too deeply into.
- nenenejej 1y agoIf that is the case the counterexample is the sort of thing a stubborn cynical and amateur mathematician (and programmer) may have found.
- exabrial 1y agoAll I know is a triple fisherman's is nearly impossible to untie in 5.6mm UHWMPE after taking a whip on a sling made out of it. It's sort of comforting have the rock hard knot; it'll break the cordelette before untying. Interestingly, an unweighted one is pretty simple to untie!
- rkomorn 1y agoIt's not often I come across a comment that I wouldn't understand any less if it was in a language I don't speak but I think this one makes the cut. Holy topic-specific terminology, Batman!
- Etheryte 1y agoRelated: "How I, a non-developer, read the tutorial you, a developer, wrote for me, a beginner" [0]. [0] https://news.ycombinator.com/item?id=45328247 https://news.ycombinator.com/item?id=45328247
- rkomorn 1y agoHaha, yes!
- jkingsman 1y ago> All I know is a [knot used to join a length of rope into a circle/loop, performed three times) is nearly impossible to untie [when tied with large-diameter polyethylene plastic rope] after [falling (while climbing) and being caught by the loop I made to take load] made out of it. It's sort of comforting have the rock hard knot; it'll break the [loop itself, structurally] before untying. Interestingly, [if you don't tighten the knot by dropping bodyweight from a height on it like I did, they're] pretty simple to untie!
- crazygringo 1y agoThis is amazing. Thank you!
- 1y ago
- rao-v 1y agoIt does suggest that there are many important problems out there that are amenable to relatively cheap brute force search at this point.
- FungalRaincloud 1y agoA lovely thing in math is that a counterexample, especially if it leads to infinitely more counterexamples in a particular class, can teach you more about the problem. I find this article hopeful. It made me excited about knot theory for the first time in a while.
- 1024core 1y agoSpeaking of knots, and not to hijack this post: I am interested in learning, say, 10 most useful knots that could be useful in most situations: joining two ropes, attaching a rope to a tree branch, etc. etc. Is there a youtube channel people would recommend I watch to pick them up?
- greenicon 1y agoI'd say that the basic sailing knots should fit your bill pretty well. I can't recommend an online source, but you should find plenty resources on Youtube. It shouldn't take longer than an evening or two to learn them.
- iljya 1y agoAnimated Knots by Grog is a good reference, with excellent explanations, visuals, and in many cases fascinating supplementary commentary and history. Have fun!
- alex989 1y agoI used to be a rope access worker, mostly for consstruction, maintenance and inspections in hard to access places. Most knots are only useful in very niche situations or to impress your friends. You probably don't need more than 5 to solve almost every situation you could realistically get yourself into (eg. Figure height, alpine butterfly). In a lot of cases, the fancy knots you see online are only usefull because they are easier to untie after getting loaded (eg. Using figure-nine instead of figure-height) and you can ignore them. I would recommend looking at the ones that are thought in the Irata and Sprat certifications. IIRC there is fewer than 10 but there is a wide range of ways you can use them or combine them together.
- bubblyworld 1y agoFigure eight, bowline, slipknot, clove hitch, trucker's hitch (not really a knot but useful) and a sheet bend will cover you for like 99% of use-cases, including climbing haha.
- justusthane 1y ago
- anigbrowl 1y agoEvery Quanta story posted here seems to be 'simple math thing is unexpectedly difficult' or 'elegant solution to difficult math problem is unexpectedly simple.'
- NooneAtAll3 1y agoisn't that basically most science journalism?
- paulgerhardt 1y agoI’m sure I’m missing something here but isn’t this common knowledge in sailing and climbing? Specifically tying a knot with opposite chirality to one existing on a line can cause both knots to capsize and roll out. One would not take it as given that three knots plus three knots would yield six knots in this scenario.
- 3cKU 1y agoIt does seem the obvious place to start looking, and the only surprise is that it took so long.
- thomasahle 1y ago> The duo kept their program running in the background for over a decade Were they not searching for counter examples to the sum conjecture during this time? Or how did they program not identify this simple 3+3 example sooner?