All 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!
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
Do you by chance mean 5kn, not 14kn? I believe thats the threshold of injury. I believe > 8kn is pretty much certain death, but I don't have a direct source.
> a double-figure-8 (the standard rope<->harness knot) will break at around 14kn
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).
The double-figure-8 is pretty rare in climbing, it's the regular figure-8 loop that's used. Sometimes called a "figure-8 follow-through" to describe the tying method. The "double-figure-8" has two loops, it's not at all commonly used in climbing. Mostly just for some improvised rescue situations as part of an improvised harness to replace a damaged harness's leg loops.
> 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!
Speaking 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?
Figure 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.
If you learn some of these, you'll also see how interconnected many of them are.
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.
I 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.
I'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.
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.
I work as a merchant seaman and for our regular day's work everyone basically exclusively uses bowlines, round turn and two half hitches, and clove hitches. We'd use reef knots or single sheet bends for joining ropes.
Animated Knots by Grog is a good reference, with excellent explanations, visuals, and in many cases fascinating supplementary commentary and history. Have fun!
I settled on two, I don't really need a lot of knots but wanted to do better than the overhand. Anyway the two I settled on were "putting a loop in the end of a rope" The bowline. and "tying something down on my truck" The Truckers hitch. There are many great knots out there but I figure between these two that is about 90 percent of my knot tying needs.
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.
I went through that exact rabbit hole a couple of years ago, and after much watching the HowNot2 channel, and much reading other sources, I came to the conclusion that you only need to learn very few knots to do nearly everything. The specific knots you learn matter little as long as you select knots with a long eatablished safety record.
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.
> you only need to learn very few knots to do nearly everything
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.
> 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.
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?
> 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.
To be frank, this has been a rather disappointing conversation.
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.
The problem is that it's nearly impossible to remember the correct way to tie a specific knot in a given situation unless you do it frequently. It's always best to have a cheat sheet as part of your (e.g., camping) gear. So once, you get the YT recommendations that you ask for (and you will), take physical notes!
I keep a short length of rope around the house where I occasionally pick it up and practice some knots. If you rely on them to protect yourself like in climbing they needs to be memorized to the point where you can tie them in the dark.
There are kits you can buy which come with a deck of cards along with two or more ropes (of different sizes) which are fun to do as well. Some knots are specific to connect different size or same sized ropes. It's a good mental exercise.
There are plenty of good examples in the other replies and I just want to add that the square knot, with a tiny variation, is the best way to tie your shoelaces.
> 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?
Maybe 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"
Yup. I've had lots of intuitions for things, only to discover there was a very non-intuitive theorem conclusively proving my intuition wrong.
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.
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.
Maybe 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()?
An 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.
I 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.
That 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.
Every Quanta story posted here seems to be 'simple math thing is unexpectedly difficult' or 'elegant solution to difficult math problem is unexpectedly simple.'
A 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.
All 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!
HowKnot2 have been making empirical testing videos on climbing knots and gear which have proved fascinating and often unintuitive.
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
Do you by chance mean 5kn, not 14kn? I believe thats the threshold of injury. I believe > 8kn is pretty much certain death, but I don't have a direct source.
Actual tests:
* https://web.archive.org/web/20250712222155/https://www.howno...
* https://web.archive.org/web/20240125125133/https://www.howno...
The anchor experiences greater force in a fall than the climber.
Ah I see now, "force experienced by rope" not "force experienced by climber" :)
> a double-figure-8 (the standard rope<->harness knot) will break at around 14kn
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).
The double-figure-8 is pretty rare in climbing, it's the regular figure-8 loop that's used. Sometimes called a "figure-8 follow-through" to describe the tying method. The "double-figure-8" has two loops, it's not at all commonly used in climbing. Mostly just for some improvised rescue situations as part of an improvised harness to replace a damaged harness's leg loops.
yup, it's also great for fixing and equalizing a rope to a sport anchor for top rope soloing, etc. https://www.alpinesavvy.com/blog/fixing-a-rope-two-knots-to-.... Super easy to adjust/equalize.
Yep, you're totally right. I had double and triple on the brain from the Fisherman's; standard climbing knot is but a simple figure-8.
It'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!
> 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!
This is amazing.
Thank you!
Related: "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
Haha, yes!
All we need is a fisherman-toplogist, and we can perfect the incomprehensibility of the discussion on particular knots.
This may help.
https://m.youtube.com/watch?v=TUHgGK-tImY
Or may not.
You probably know this already if you climb, but in east germany and the czech many areas mandate knots as protection, jamming them in cracks: https://www.climbing.com/travel/soft-stone-rigid-ethics-elbe...
Supposedly cams and nuts damage the rock. Pretty gnarly stuff. And it's often sandy off-widths as well...
Speaking 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?
Figure 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.
If you learn some of these, you'll also see how interconnected many of them are.
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.
I’d add the adjustable grip hitch to your (very good) list.
A round-turn and two half-hitches is also useful.
And a highwayman's hitch, just for fun
I rarely use anything other than a bowline, midshipman's hitch (and variants, very useful) or zeppelin bend.
A reef bend also works, but has many ways to tie it wrong.
The most important thing is to know when to use what.
I 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.
I'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.
There are three knots I use that have covered all my needs as a casual camper:
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.
PSA: the square knot is notoriously unsafe and should never be used in safety critical applications. Ditto for the sheet bend.
If you care about safety, look for knots used in climbing instead.
And the bowline is considerably weaker than the figure-8. The latter is also easier to learn and remember.
I work as a merchant seaman and for our regular day's work everyone basically exclusively uses bowlines, round turn and two half hitches, and clove hitches. We'd use reef knots or single sheet bends for joining ropes.
As a relatively keen sailboat racer, that sounds about right.
Animated Knots by Grog is a good reference, with excellent explanations, visuals, and in many cases fascinating supplementary commentary and history. Have fun!
I settled on two, I don't really need a lot of knots but wanted to do better than the overhand. Anyway the two I settled on were "putting a loop in the end of a rope" The bowline. and "tying something down on my truck" The Truckers hitch. There are many great knots out there but I figure between these two that is about 90 percent of my knot tying needs.
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.
https://www.youtube.com/watch?v=1J8MuOWO0Qs
https://www.netknots.com/rope_knots/top-10-rope-knots
I went through that exact rabbit hole a couple of years ago, and after much watching the HowNot2 channel, and much reading other sources, I came to the conclusion that you only need to learn very few knots to do nearly everything. The specific knots you learn matter little as long as you select knots with a long eatablished safety record.
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.
> you only need to learn very few knots to do nearly everything
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.
> 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.
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?
> 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.
To be frank, this has been a rather disappointing conversation.
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.
The problem is that it's nearly impossible to remember the correct way to tie a specific knot in a given situation unless you do it frequently. It's always best to have a cheat sheet as part of your (e.g., camping) gear. So once, you get the YT recommendations that you ask for (and you will), take physical notes!
I keep a short length of rope around the house where I occasionally pick it up and practice some knots. If you rely on them to protect yourself like in climbing they needs to be memorized to the point where you can tie them in the dark.
There are kits you can buy which come with a deck of cards along with two or more ropes (of different sizes) which are fun to do as well. Some knots are specific to connect different size or same sized ropes. It's a good mental exercise.
There are plenty of good examples in the other replies and I just want to add that the square knot, with a tiny variation, is the best way to tie your shoelaces.
> 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?
Maybe 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"
And yet it almost always works. There were no known counterexamples where it failed until this was published.
Yup. I've had lots of intuitions for things, only to discover there was a very non-intuitive theorem conclusively proving my intuition wrong.
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.
I'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.
Mine too, but this is not very interesting, although our intuition was right, it was almost certainly right for the wrong reasons.
Agreed, this is the real takeaway for those who think it’s unsurprising: they simply haven’t understood why it is surprising.
Definitely!
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.
If that is the case the counterexample is the sort of thing a stubborn cynical and amateur mathematician (and programmer) may have found.
Maybe 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()?
An 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.
I 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.
That 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.
Not exactly. Orientation is a bitch, just make a mobius strip anc cut it across its middle, and do it again… magic.
I 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
I’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.
It does seem the obvious place to start looking, and the only surprise is that it took so long.
Every Quanta story posted here seems to be 'simple math thing is unexpectedly difficult' or 'elegant solution to difficult math problem is unexpectedly simple.'
isn't that basically most science journalism?
It does suggest that there are many important problems out there that are amenable to relatively cheap brute force search at this point.
A 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.