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
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?
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.
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 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.
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.
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.
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 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.
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.
Animated Knots by Grog is a good reference, with excellent explanations, visuals, and in many cases fascinating supplementary commentary and history. Have fun!
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.
Every Quanta story posted here seems to be 'simple math thing is unexpectedly difficult' or 'elegant solution to difficult math problem is unexpectedly simple.'
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.
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.
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
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.
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?
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
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.
https://www.netknots.com/rope_knots/top-10-rope-knots
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.
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.
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.
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 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.
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.
Animated Knots by Grog is a good reference, with excellent explanations, visuals, and in many cases fascinating supplementary commentary and history. Have fun!
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.
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?
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.
Not exactly. Orientation is a bitch, just make a mobius strip anc cut it across its middle, and do it again… magic.
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.
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
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.
It does suggest that there are many important problems out there that are amenable to relatively cheap brute force search at this point.