A big problem we get with the eyes we have been born with, is that we can not see “forces” ! We see the shape of a structure, but not the “flow” of tensile forces, the “concentration” of compression forces, the “accumulation” of torsion within it… This makes us make biiig mistakes - it is nothing but the equivalent of the fact that " Appearances/looks are deceptive " in the field of structures and knots. A dome and a fishing net can have the same shape, but in the former the structural elements are withstanding compression, while in the later they are withstanding tension. However, we can not see, even that - and we have to decide which structure is what, only by taking into account other experiences.
A knot is a complex structure, where we have tension, compression and torsion at different points, at the same time. However, many knots are very stable structures, so their form does not change drastically when the distribution of all those “forces” vary at various points of the knot. The shape of the fig.8 bend, for example, will not change drastically, even when it is loaded by the one or by the other pair of ends, so we tend to think of the two knots as the “same” knot. However, if we could see the “forces” inside the rope, if we could watch their “flow”, accumulation and dissipation, we would have seen at once that they are distributed very differently, and so those two knots “work” differently - even if, at the end, they both “do” the same job they have to do : they secure their tails. They do what they do differently, they “work” differently, different parts of them are tensioned and compressed, at different points, so one could claim that they are different knots.
When I was a young student, we had a teacher who was always trying to show to us how a structural element “feels” a certain “force”, by some so funny movements of his body, hands and feet, that make me laugh till now. He was always telling us to “feel” the forces throughout the structure, before we start to calculate them. In a complex structure, sometimes you have to use all your imagination to figure out if a certain element is tensioned or compressed !
Now, I try to do exactly the same thing in the case of knots : I try to “see” what runs “inside” the rope, where the rope “feels” tension and compression and torsion ( the later is the MOST difficult thing ! ). When I do this, I “see” a knot not only as it looks and can be seen as shape, but as it works and can be analysed as structure. Different structures are different knots - the “reversed” of a knot may be similar or identical, as a structure, to the initial knot, or not.
( In the case of the Mobius loop, for example, we do not even have the “reversed” of the Butterfly loop - because the in-line Butterfly loop is NOT “working” as the end-of-line “Interlinked Englishman s knot”, which is topologically the same, but geometrically veeery different, so it also “works” differently. The “reversed” Butterfly loop is meant to be loaded as the Butterfly loop itself, as an inline knot, NOT as the Mobius s loop. I do not believe that this knot is stable, and it may even be proved, by tests, that the “reversed” of the Butterfly loop is only the Butterfly loop ! - and all the other “intermediate” forms may be proved to be unstable, when loaded by ends pointing to opposite directions.
On the other hand, I think that Mobius s loop is stable, when loaded as Mobius shows, i.e., by its ends when they are pointing to the same direction, opposite to the direction its eyelegs point to. However, I had NOT loaded it as heavily as I should, so I really can not tell that it is a stable knot. )
To answer your question: “The shape of a knot after dressing-before loading : does it determine the knot”? I think, yes it does. As it turns out I was thinking about exactly this issue after your comment about the Mobius Loop now being a “Reversed Butterfly Loop” after I trialled the Mobius Loop end-2-end loaded with no load on the loop.
The particular question raised by xarax revolves around the naming of knots and the discussion about what is new and what is not. Let me consider that related issue for a moment. Mainly using the Mobius Loop as an example of different viewpoints:
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If two knots are topologically equivalent then both knots are ‘the same’, especially if one knot is deemed to easily capsize to the other. Therefore there is not a good reason to use the name “Mobius Loop” when it is the same topological knot as a Butterfly Loop, and it can be capsized to Butterfly Loop. The same viewpoint was expressed about the Farmer’s Loop and the Plait Loop recently, so this idea is not without supporters.
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Two knots may be topologically equivalent, however the geometrical structure of the two knots are different enough to name them as individuals. Note that there is no mention of how the knot is to be loaded, it’s just about how it is shaped. This, I believe, is the primary reason the Mobius Loop got it’s name as an individual.
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Two knots may be topologically and geometrically equivalent though we differentiate between them (and name them separately) because of the way we load them. As an example, the Mobius Loop was no longer the Mobius Loop and becomes a Reversed Butterfly Loop (as expressed by xarax) because of the loading I placed on the Mobius Loop structure.
I find viewpoint 1) and viewpoint 2) quite reasonable, and I certainly internally debated the two ideas and expressed some that at the beginning this post: http://igkt.net/sm/index.php?topic=5315.msg35407#new
I do not find 3) very helpful at all. It reminds my of the story “The Cat in the Hat comes back” I read to my children the other day. The cat has a hat, under his hat is ‘little cat A’, under little cat A’s hat is … you guessed it little cat B… and so on ;D The point being is that you think you have tied a well known knot (the Cat) and then you go to load it a bit differently than normal and someone tells you “that’s not the Cat that’s little cat A” … say what?
If I’m wrong about option 3) then the Devil might play a trick on me later: He might say, “mobius, tie a Butterfly loop and I’ll let you out of here”. I get very excited and tie my Butterfly Loop. Then the Devil says, “Ok mobius, place the eye over one of my horns and pull hard”. Still excited, I do that and think, “I’m out of here, yippee”. However, the Devil now says, “bad luck mobius, your knot isn’t a Butterfly Loop, you only loaded it from one end”.
Had a bit of fun with my writing, however I hope I made my point ;D
Cheers,
mobius
[Edit: sentence struck out due to a misinterpretation on my part]
2) Two knots may be topologically equivalent, however the geometrical structure of the two knots are different enough to name them as individuals. Note that there is no mention of how the knot is to be loaded, it's just about how it is shaped. This, I believe, is the primary reason the Mobius Loop got it's name as an individual.
Viewpoint 1 should be pretty uncontroversial especially in cases where capsizing is very unlikely.
I agree that viewpoint 3 is not very useful. There is a point where we just have to talk about how “things” are used.
But I disagree with your characterization of 2 in regards to the Mobius loop. There is no difference at all in the geometrical structure of the knot nub at least. The difference ironically is strongly related to item three, even though I disagree with three by itself but support 2 and even the mobius version of 2 (if stated correctly). The difference in the mobius B. LOOP rope structure (the word knot really is what causes trouble and is simply too limited) is not about how it is loaded or about the structure of the knot part but about the relationship of the knot part to the rope, creating different potentials for how to load it under different uses. THAT is a significant difference. The mobius B. Loop rope structure is a different rope structure but saying the knot has a different geometrical structure when unloaded depends on what one means by the geometrical structure of a “knot” especially in the context of not considering loading.
Indeed simply saying you tied an inline loop and hung on it, and it was a mobius, and then you hung something from the tail and now it’s not, that’s getting absurd and doesn’t fit well with natural distinctions between what a thing is and how a thing works in different conditions.
Probably the biggest reason why (3) is unreasonable is because it is uneeded. We use different words because we need to communicate something that the first word does not and in way that simplifies our communication. We do not need a different word to communicate an end-to-end loaded mobius. B. Loop and adding a different word does not improve or simplify communication. Yes it saves writing “end-to-end” loop but also looses the clarity to unfamiliar readers that writing it out adds anyway. I see no linguistic utility in having a different word, only linguistic baggage such as needing to change words while describe the same part of a system under changing conditions. Switching words mid stream like that can only add to communication confusion, not reduce it.
It’s not entirely clear to me that anyone really seriously advocated for option 3 when push came to shove. I think xarax was trying to explain something similar to my view of option 2 but struggling to find it and instead described 3 in slightly miss-aimed effort. When he actually did say to call the end-to-end version a different name, he did include a smiley and I give him the doubt that he was at least somewhat jesting, ie not really expecting it to reasonably become a strictly observed linguistic distinction on a regular basis. Rather maybe he was just using this exagerated linguistic distinction to make some point that you should not be too worried about how the Mobius B. loop works in those conditions.
I took the liberty of adding the capsizing element to my previous post, though that now introduces an element of loading being important.
But I disagree with your characterization of 2 in regards to the Mobius loop. There is no difference at all in the geometrical structure of the knot nub at least. The difference ironically is strongly related to item three, even though I disagree with three by itself but support 2 and even the mobius version of 2 (if stated correctly). The difference in the mobius B. LOOP rope structure (the word knot really is what causes trouble and is simply too limited) is not about how it is loaded or about the structure of the knot part but about the relationship of the knot part to the rope, creating different potentials for how to load it under different uses. THAT is a significant difference. The mobius B. Loop rope structure is a different rope structure but saying the knot has a different geometrical structure when unloaded depends on what one means by the geometrical structure of a "knot" especially in the context of not considering loading.
I too think we need not overstate the loading argument aspect. We can just think about the geometry in terms of naming parts of the nub: Spart, Tail, two eye legs. If the two eye legs become Spart and Tail: because we joined the ends, cut the loop and then redressed the nub (even if that is only very slight), I think we have done enough to call the new form a new knot.
Have at look at 1) again and help me reword 2) if you wish 
Cheers,
mobius
You could well be right, though I certainly didn’t take it that way when I read the following in another thread…
“THIS is not your knot !” was referring to my trial of the Mobius Loop. xarax will soon tell us what he thinks and hopefully clarify things, I’m sure 
Cheers,
mobius
directed at mobius
xarax I agreed with many of the things you said here. Anyway, this point is just wrong, condescending, marginalizing and discrediting to your point. Mobius clearly understands the differences in loading, certainly at least that there ARE differences and that they are very important, important to how the know behaves and even if it survives as a knot. Nobody has denied this.
You are now trying to marginalize opposing opinions about a purely subjective matter by repeatedly trying to pretend as if the person holding that opinion simply does not understand the situation as well as you do in ways where it simply clearly is not the case. This is easier for you than having an actual discussion where you need to consider that not every well informed person will agree with you, but it doesn’t get anywhere. The fact is there is no right and wrong answer to such things. Your science argument doesn’t hold water under scrutiny and I would conjecture that sentiment 3) fails miserably to find support among general population. Joe trucker is not going to tie a Mobius knot and suddenly call it a different name because he then decides to tie the tail to something else.
Again what makes it different is that if you DO ring load the loops the same way (for example) , it results in loading the knot part a different way for the two. So the relationship between the knot behavior and the knot interface to the world, is not the same. That doesn’t mean I can’t find loadings that stress the knot part the same, but I have to use the knot differently to do it. It also doesn’t mean we need to name every loading with a different knot name.
Xarax what would you call a ring loaded alpine butterfly loop? I’m sure this configuration must have been discussed before. Did you never feel the need to give it a new knot name?
Knots that are topologically identical, may be geometrically very different, have different structures and structural characteristics regarding slippage, strength and jamming, “work” differently, and so be different knots. One can go as far as to consider topologically and geometrically identical knots as different, if/when, without any major change in their forms, they are loaded differently. A loop may retain its basic shape when it is loaded by the one or the other free end, but is it the same knot, or it is two different knots, which, when loaded, just happen to have a similar shape ?
However, knots topologically different, may have similar geometrical forms and similar structures, “work” the same way, and so be essentially the same knot…( Or not ! To this day, I can not make up my mind and decide if the Carrick bend and the Carrick X bend ( X=crossed Tails ) shown in (1), should be considered the same knot, or not…)
So, let us not pay too much attention to the definitions and the names of knots - we have just ro remember that a knot is a rope-made friction machine, and when a particular knot works differently than another, it is a different machine, so a different knot, no matter of how it looks - and vice versa : if two knots look different, yet “work” the same way, they may be just two variations of the same knot.
Of course, we are not talking here about knots which have different forms, yet they can easily capsize into the same knot. We are talking about STABLE knots, knots which retain their geometry, and so their structure and the way they “work”, even during and after heavy loading. One can argue that, under heavy loading ( or, for that matter, under ANY loading ) the geometry changes a little, because the knots shrinks, and some parts of it shrink more than some others. However, I think that it is a much easier thing to distinguish between two different knots, and two slightly different shapes of the same knot when it is loaded lightly and when it is loaded heavily.
I have also another thing to mention : “re-dressing” is a vague concept : “redressing” may keep topology the same, and, in most cases, it does, indeed - but it can also change it, as when happens when we redress the final segment of the Tail Ends, before they leave the nub, as it happens in the case of the Carrick and the Carrick X bend(s). To my view, redressed knots may be essentially the same or they may be altogether different knots, and it may be not-so-easy to go from the one to the other by manipulating the segments of the nubs - as it happens with the Scot s TIB bowline and the Ampersand TIB bowline, for example.
With the one knot, when the tension descends from the Standing end it is nipped here so much, and there so much, and it is turned into friction here so much and there so much, and that is why there will be no tension left at the Tail End. With the other knot, all of the above, but with different “here” and different “there” ! Two different rope-made friction machines.
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If the geometry / shape of the knot does not change, but only some parts of it shrink more than others, the knots are not different.
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ALL knots shift their geometry to accommodate different loadings ! Segments are tensioned and elongated here more than there, curves get sharper, collars rotate, etc. In fact, it is amusing to watch what the poor knot has to do, to bear the load !
However, the difference between a stable knot which just “accommodates” loading, and a knot which capsizes under loading, is not a grey thing, it is a black or white thing. I do not believe there is gradual capsizing ; When it starts, it will go straight to the end, and will generate a completely different form. -
Different loaded ends, different structures, different knots. Shape is NOT the only thing it matters - and it may even be different, and yet the structure be essentially the same .
“Capsizing” does NOT mean “redressing” ! “Capsizing” is made by the load, NOT by the knot tyers fingers !
If, when a knot is loaded, and without any intervention by the knot tyer, its initial, compact shape is transformed to a completely different shape, then, and only then, the knot has capsized. If the shape of the knot changes when the knot tyer change it, the only thing we can conclude is that the rope itself and the knot tyer s fingers do exist, they are not petrified and they are not frozen.So, in order to see if a knot capsizes to another knot ( i.e., if it is stable or not ), we have to load it. One would argue that many knots, perhaps most of the knots, under extremely heavy loading do slip or capsize ( which, regarding the abrupt and complete change of the initial form, are the same thing ) - therefore we have to define this notion more carefully. However, it is simply wrong or meaningless to say that when, by manipulating the segments of an unloaded knot, we change its shape, the knot “capsizes”.
The “half hitch” and the “nipping loop” may be considered as having identical “parts”, but they are not the same. If “difference in dressing” means “difference in shape”, without any consideration of what is going to be loaded, and how, I think this “difference” is not what makes knots be the same or be different ! Structures are not determined uniquely by their shape. A dome and a fishing net may have the same shape, but they are altogether different structures.
I would like to say something more about the difference between “capsizing” and “redressing”.
To me, “capsizing” is related to a spectacular, in space, and abrupt, in time, transformation. I would compare it to a “catastrophic” event, where we have “sudden and dramatic changes” (2).
" Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. "
" A knot that has capsized has deformed into a different structure" (3).However, by this deformation we should not be mean the result of the manipulation of the knot tyer s fingers ! We can always change the structure of a knot by manipulating its segments when it is unloaded, and that is why we are able to tie knots on the straight unloaded line ! This deformation should be the result of loading of the knot - not of tying or dressing the knot. The deformation which is the result of a “re-dressing” is voluntary and gradual / smooth, the deformation which is the result of a “capsizing” is involuntary and abrupt / spasmodic.
A knot is “stable” when it does not capsizes to another knot - at least when it is not loaded in the extreme, near its strength limit ( However, there is an issue there, because some “stable” knots, do capsize under extremely heavy loading, but we still consider them as “stable” ). It would be meaningless to say that a knot is 'stable" when it can not be re-dressed in an unstable form ! -
http://en.wikipedia.org/wiki/List_of_knot_terminology#Capsizing
?! How can one say this when e.g. “2” directly shows how
it is so false --that “topological equivalence” e.g. makes all
TIB knots = no-knot!?
Moreover, I don’t agree that equal-via-capsizing is “same”
(e.g., the bowline capsizes into a pile-hitch noose).
A look at the recently posted video(s) of a fig.8 eyeknot loaded
to rupture also sheds light on questions of geometry --of what
was then was not --in that material-- in terms of angles of
deflection, compression & friction of parts, and so on.
A question is : What is the use of this debated “sameness”/“equivalence”?
–to decide claims of New Knot, or to articulate physical behavior(s),
or … ?
But, wait, we are past the fundamental question :
What is a knot?
IMO, its an entanglement of cordage WITH specific loading.
(So, a bowline/(a/b/c/d) / sheet bend-a / sheet bend-b/ net knot / becket hitch
though looking --by helping hand of artist-- identical in the
cookie-cutter/all-ends-run-out-of-image-circle view (of the
“entanglement”), they are differentiated per loading (which
has greater & lesser effects on respective geometries).
–dl* (Who thought he had VERY “same” URLs, elsewhere … !!)
DL I think you read my comment on point 1 in the reverse of how I meant it.
However, I will take issue with one point you said. I am not amused by saying that TIB knots are equivalent to the unknot, a tired and useless argument, even as a straw argument to be countered, in my opinion. The mathematical notions of knots, knots should have no free ends either. If you can untie a knot then it’s topologically equivalent to an unknot in the strictest way of thinking. Oh am I being too.. technical ? Well that’s the point isn’t it. Brushing this off can even cause some minor conundrums. If you start transforming between coordinate systems with different curvatures it might even be difficult to define when a knot is tied or un-tied if it is allowed to have ends. Ok, ok, there is still an easy fix, for real knotting purposes we just need to imagine that the ends go off to infinity or that are affixed to a surface, an infinite plane (like that they are attached to post that itself connects to the ground, and we assume that we cannot pull bights around the earth because it is effectively infinite). This can solve the problem without crumbling down our entire notion of topology for real knots.
Fine, but if we’re doing that we can also fix up loops by imagining the loop legs go off to infinity or, maybe better, that the tip of the loop is attached to an infinite plane that cannot be violated, or that an infinitely long post (or simply an ring affixed to the post in the earth) extends through the loop. If loops are critical to defining loop knots then we cannot say two loops are topologically equivalent if they cannot be transformed while used as a loop knot. I find this “fix” no more arbitrary or contrived at all than the one for needed for the end of the rope.
updated:
In summary I think this unknot TIB argument is actually quiet useless even as a straw argument. One does not need to see topological equivalence between all TIB loops and the unknot to see some topological equivalence between two TIB loops. In in a technical sense it is not hard to find an easy way to distinguish the one significant topological similarity from the other absurd one. If the anti-straw argument was the only reason I agreed with you about point 1, then I certainly wouldn’t agree with you about point 1. The real reasons are not as satisfyingly provable and there are probably cases where it is not so satisfyingly clear.
I updated my last post to make clear that I understand the TIB unknot point was a straw argument and find it failing even as such, in precisely the way that many straw arguments are for that matter.
If I understand you correctly, I don’t mind the a/b/c/d idea as it makes naming knots based on shape and load a much more useful approach. If I may use your EBDB Bowline as an example of name/load differences:
EBDB Bowline - implies the normal standing part (SPart) and eye loading of the knot only
EBDB (T) Bowline - implies the same knot shape as above, however someone has looked at loading the tail
EBDB (R) Bowline - ring loaded version
EBDB (E) Bowline - end to end loaded version
EBDB (4) Bowline - end to end and eye loaded (could be 3 instead of 4 I suppose) version
.
etc if there are more ways to load.
I realize that for many knots we are not interested in any loading except the standard one. However, with some of the Either End Loaded (EEL) loops we are, so having a suffix (a/b/c/d) or those I just made up (or something similar) might be useful.
Not losing the parent knot’s name and adding a suffix to accommodate loading differences makes sense to me and I think this is essentially what you are saying above.
Cheers,
mobius
The Munter hitch / crossing knot is an interesting form that can be loaded on one strand or the other, transforming itself in the process, but nevertheless remain stable.
Edited: Fixed typo error (one strand or the other)
Wow, that one was not obvious, I’m not really sure how I did it, because my most consciously understood efforts were failing. I was trying to get the Scott from the hitch/noose redress of the ampersand (which is surprisingly convoluted for being born of a pretty simple bowline retuck) but that wasn’t working out. And then some last ditch twisted thought in the back of my brain just made it happen but I think not by chance. And, now and forever, the first Scott Bowline I ever tied, was using this method.
Hmmm… We can say that it “remains stable”, only if we ignore the carabiner ! I mean, the carabiner, the ring, or the bight within which it is tied, should also be taken into account - and when we will do this, we will see that the “nub” of the hitch has changed the side of the carabiner or the ring or the bight/eye within which it was tied initially, so it has been “transformed” into “another” hitch, “another” knot, which now looks the same, and works the same way… Or, it has only been “transported” to the other side of the carabiner or the ring or the bight, and it has also been “reversed” at the same time.
Anyway, the Munter hitch is a mechanism, rather than a knot. The nub of the knot is a half-of-a-Cow hitch, or a half-of-a-Clove hitch ( the latter is how it is seen, traditionally ) - and the half of the Cow hitch or the half of Clove hitch do not transform themselves.
No xarax, that was not from another thread, it was a direct response to your own post two posts up.
Indeed. It took me ages to see it, however the Bowline I have bee working on does have a Munter hitch shape to it’s turNip. It was hard to see since the tying method doesn’t use a Munter hitch and depending on how the loose knot sits it can look like a ‘pretzel’. I have done quite a few trials on it and will share with everyone in due course.
Cheers,
mobius
You mentioned the Scott and ampersand bowlines as an example of topologocial equivalents. My post was a tangent, and I probably should have quoted yours, but it was a tangent related to the discussion, and this is a tangent of that. Anyway, it was a fun and certainly not obvious (to me) thing to see, so thanks for sharing that.
No, I have never expressed the opinion that, because the Farmer’s loop and the Plait loop can be transformed topologically one into the other, they are the same . That’s a misinterpretation of my statement “Probably better used in it’s ABoK #1054 Farmer’s Loop form”. I made that statement for two reasons. 1) to see if anyone, including xarax, had noticed the relationship between the two forms, and 2) suggest that IMO the Plait loop has no advantage over the Farmer’s loop.
Even now, your language is NOT precise, to my knowledge of this language... What does this " solely " means ? I can read it as " They ARE the same, but not only because..." ( meaning that this is not the sole / only reason, there are also other reasons for this "same-ness" ). The sentence I would had understood, is : "The Farmer s loop and the Plait loop are not the same, although they can be transformed topologically the one into the other ".I see your point and have edited above.
I have a hard time reading the page-long writings of Dan LehmanYou and me both. ;D
I understand that you have not understood anything of my long post, with the MANY advantages, IMO, I had described / explained, a few days ago - and I will not repeat my arguments. Simply I have to point out that you should not compare two different things, an in-line loop to an end-of-line loop. If you want an in-line loop, stick to the great Butterfly loop. I wanted something else...Not true. I've read, and understood, [b]all [/b]of your arguments and thanks for not repeating them. Who says the Farmer's loop is only a inline[sic] loop and not an end-of-line loop? The Parrot parroting the parroters? :)