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.
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.
So much of math and physics is discovering these beautiful, surprisingly non-intuitive things.
And this fits right in that pattern -- it seems intuitive that it wouldn't be true, but nobody's been able to find a counterexample. So it's yet another counterintuitive result that math is built on. Not proven, but statistically robust.
Which is what makes it great when somebody does ten years of work in simulating knots so a counterexample can be found.
Which doesn't even confirm the original intuition, because there are still so many cases where the rule holds. Whereas our intuition would have assumed a counterexample would have been easy to find, and it wasn't.
Holy topic-specific terminology, Batman!
Thank you!
https://m.youtube.com/watch?v=TUHgGK-tImY
Or may not.
If you'd like to see a break-strength test comparing single, double and triple fisherman’s knots, you might enjoy: https://www.youtube.com/watch?v=5CAjUi47QMY
For reference, a 100kg climber is unlikely to be able to cause more than 14kn of force on a dynamic rope, (in reality, significantly less,) even if they go out of their way to find the worst case fall-scenario. Most belay loops are rated at 15kn, human bodies start breaking at 8-12kn and HowKnot2 says that a double-figure-8 (the standard rope<->harness knot) all break at around 14kn https://www.youtube.com/watch?v=g4CVFRE0pRg&t=500s
Actual tests:
* https://web.archive.org/web/20250712222155/https://www.howno...
* https://web.archive.org/web/20240125125133/https://www.howno...
My understanding (based on some training aeons ago) is that the figure 8 knots prevalence for tying in is not due to that strength - but instead because it is easy to check, hard to back through/untie during usage, and strong when mis-tied (ie errors are made and not caught).
Supposedly cams and nuts damage the rock. Pretty gnarly stuff. And it's often sandy off-widths as well...
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.
A sheet bend and bowline knot are both wildly useful. But a bowline is just a single rope sheet-bent (sheet bended?) back onto itself! And a trucker's hitch is just a slip knot where you creatively use the slipped loop as a pulley.
And a highwayman's hitch, just for fun
A reef bend also works, but has many ways to tie it wrong.
The most important thing is to know when to use what.
Taut-line hitch
Sliding knot for fastening loads or setting guy-lines https://www.netknots.com/rope_knots/tautline-hitch
Bowline
Essential general knot for tying a loop. https://www.netknots.com/rope_knots/bowline
Square Knot
Used for joining ropes or just an easy to unite knot. For joining ropes you could do a sheet bend which is stronger https://www.netknots.com/rope_knots/square-knot
https://www.netknots.com/rope_knots/sheet-bend
There are a billion options so I recommend just picking a few and practicing until you can tie them quickly without references. You'll start to understand hope knots in general work and be able to pick up other knots much easier.
If you care about safety, look for knots used in climbing instead.
My personal short list are the following:
1. Joining two ropes (i.e. bend): Zeppelin bend, or Figure-8 bend. If the ropes have a very different diameter you will need a different knot, such as a Double Sheet Bend.
2. Holding on to an object (i.e. hitch): Two Round Turns & Two Half Hitches. More turns and half hitches make it more secure.
3. Making a fixed-diameter loop at the end of a rope: Figure-8 loop.
4. Making a fixed diameter loop in the middle of a rope: Alpine Butterfly, or simply take another piece of rope and do a Prusik Loop.
5. Grabbing onto a rope, such as when you want a loop that can be cinched down (i.e. friction hitch): Icicle Hitch. I personally do Round Turns & Half Hitches instead, and will die on that hill.
Another useful trick that can be done with a combination of the above is called a Trucker's Hitch. It is not so much a unique knot, but a common combination of the principles above.
For those who know about knots: please resist the temptation to nitpick and offer alternatives. Yes, there are many others. No, it doesn't matter. The knots above, or a combination thereof, covers 95% of everything you can do with rope, they are safe, and easy to verify.
I see a lot of posts here along these lines. It turns out there is a trade-off between knots: how easy they are to undo vs how likely they are to spontaneously untie, particularly when not under load.
Most of the "every knot you need" recommendations here seem to come from people tying things down for a short haul, and consequently come from the the "easy to untie" end of the spectrum. The sheepshank is great for a temporary tie down but obviously falls apart when not under load. Less obviously so does the bowline, figure 8, and most knots composed of half-hitches.
A rock climber takes a dim view of knots that spontaneously untie when they aren't looking, so they use a different set of knots. At the extreme are fishermen. A single strand of nylon is slippery, is weakened by kinks, and yet a fisherman's knot must remain secure while drifting in the surf being bashed waves. Consequently, they will use complex, slow to tie knots with 7 or 10 loops.
Your knots look to be at the "easy to untie" kind, except the alpine butterfly. If it has been under high load for a while it can be a real bitch to get apart. It's popular with climbers, but I would not recommend it for tying down a load.
Agreed. There are many tradeoffs, indeed. But just because there are ten common knots that can do a bend, it doesn't mean that a person benefits from using all of them -- they all perform the same function, so knowing a single secure bend is enough, especially for a beginner asking these sorts of questions.
Personally, I have chosen my knots based on how safe and effective they are, as well as how easy they are to remember, tie, dress, verify and untie after load. The Zeppelin bend is hard to verify against e.g. the hunter's bend or Ashley's -- but it's just as secure as a flemish bend at a fraction of the effort to tie. The double sheet bend is bleh, but I didn't want to get into the weeds of what to do when joining ropes of very dissimilar diameters.
> Most of the "every knot you need" recommendations here seem to come from people tying things down for a short haul, and consequently come from the the "easy to untie" end of the spectrum
Agreed. I would say camping-style knots tend to be easy to tie, easy to untie, and not adequate for safety critical applications.
> Your knots look to be at the "easy to untie" kind
If you mean "easy to untie after being heavily loaded", then we agree. If you mean "can become untied accidentally after e.g. intermittent loads", we disagree. They are climbing knots, after all.
I specifically did not include knots that are commonly recommended even though they untie easily e.g. under intermittent loads, such as the sheet bend or the bowline, precisely because of how easily they become untied.
> It's popular with climbers, but I would not recommend it for tying down a load.
I am not in love with the alpine butterfly variations in general, but in the specific context of making a midline fixed loop without access to either end, there's not much to choose from as far as I know. The Figure-8 capsizes in that application, for example. That said, I would rather use an accessory line with a friction hitch (e.g. Prusik loop), but an alpine butterfly is commonly used in safety critical applications as you mention, so I'm curious to learn what you would rather use in that situation.
As for fishermen and safety, how do you explain that they still commonly use the bowline or the sheet bend?
I've not seen a lot of fisherman use them myself. But if they are indeed common as you say, then it must mean most fisherman are beginners.
Fisherman are nice people. They help each other out, and they learn fast. For example when a fish falls off the line a beginner will often loudly proclaim it bit through the line. Someone with experience often takes that opportunity to look for the telltale curls in the nylon line indicating the knot had slipped, rather than being cut. If they see them, they will often congratulate the beginner on nearly catching the biggest fish of the day - it's a shame they don't know how to tie a fishing knot.
> I specifically did not include knots that are commonly recommended even though they untie easily e.g. under intermittent loads, such as the sheet bend or the bowline,
Ahh, sorry. Reading comprehension fail on my part. I confused your post with others that mentioned the bowline. It's a wonderful knot, but it must be kept under tension.
> alpine butterfly is commonly used in safety critical applications as you mention, so I'm curious to learn what you would rather use in that situation.
As I said as the start, I was thinking you were recommending knots for casual use. I'd use the Alpine Butterfly when I want something that won't slip in that situation, however I quietly curse under my breath while tying it because if it gets tight I've created a lot of work, particularly if I don't have a marlin spike handy. I haven't owned one for years now, so I go out of my way to not use the butterfly.
I gave some quick guidance for newbies with some broader context to help them, only to have somebody dismiss everything I said because they didn't pay any attention to what was written.
Next time, please do read what you reply to, especially if you are going to be dismissive. What a waste of everybody's time. I was hoping to learn something new. Bleh.
Beware the truckers hitch, it is not a real knot so should be secured by one. But super handy for making a line nice and tight. The one I picked up is this over complicated version that has the nice property that it is in-line, that is, you don't need the far end of the rope for it to work.
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.
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?
NoMoreNicksLeft•4mo ago
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•4mo ago
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•4mo ago
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
ashivkum•4mo ago
pfortuny•4mo ago
kmill•4mo ago
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.