Hitches : Are they "knots", like bends ? If they are, where are their nubs ?

[X1]

   Personally, I tend to think of "knots" as "local" rope-made machines - not as "expanded", spatially extended ones, as the Versatackle or the Trucker s hitch, for example. So, when I think of an eyeknot (loop), I imagine it as the three-loaded-ends "knot", the very nub of the eyeknot, which does not include the whole eye..
   What makes me think like this ? Two facts :
  1. There can be knots even in the absence of any friction, which rely on topology, and topology only, to be and to remain knotted. A simple overhand knot stopper, for example, loaded by both ends. When we want to clarify what a "knot" is, we should better start from the simplest cases, where the knots can be revealed in their naked, most essential form.
  2. When all the ends of one of those knots are being pulled ( and they have to be pulled, all of them at the same time, in order an already knotted knot remains knotted, otherwise the non-loaded end / tail will be swollen into the knot and then slip out of it ), what will this knot become ? A local maximally tightened tangle, a nub, that will reach a state of minimum rope length. Therefore, one has to conclude that the "ideal knots" that have lost all their "physical" properties except their mathematical ones. when they will settle in a final stage of maximum shrinking, they will become nubs - local maximally dense tangles of segments of minimal ropelength .
   That is why, when I say " knot", I mean the knot s nub, the dense part of the rope mechanism which may include many other elements, but which is concentrated at a "local" area of the space. In this view, the "knots" of the Versatackle and the Truckee s hitch are their eyeknots - meaning the very nubs of their loops - all the other are "external" knot elements, which, together with the "knots", form a compound knot, or a spatially extended rope mechanism.
   So far, so good. One would not find very difficult to agree that there are compound knots, indeed, i.e., spatially extended arrangements of knots connected through tensioned lines -  arrangements which constitute rope mechanisms, not "knots". Then, one would be ready to agree that a loaded knot will settle to a tangle of minimal ropelength, where all the excessive portion of the free ends would have been consumed, and the only thing that would be left would be nothing more than the "knot" itself = the nub...until -
  - until hitches pop out !    What are those rope mechanisms ? Are they "knots", or compound knots, spatially extended, non-local rope mechanisms arranged around a pole or a main line ? If they are but "extended" rope mechanisms - rope tangles plus tensioned segments of ropes in between them -, where are the "knots" of those mechanisms ? One would show me a hitch and tell me : Show me your "knot" s nub ! What am I going to respond, in the case of a Clove hitch, or a Cow hitch, for example ? Is  the main line or the pole an important, functioning part of the knot, because it does not allow it to degenerate into the unknot and become a straight line, or it is only a neutral, non-functioning element, a part of the rope mechanism ?
   A related question can be this : Is the main line or the pole that penetrates the compound knot, a part of it ? There are many cases where, if we remove this main line or pole, the hitch will collapse to a nub, and other cases when it will simply disappear - the Constrictor, for example, and all the other TIB hitches.
  This question has been addressed in the past (1), for the hitches as well as the binders, but was not answered . Now, with this amusing game of the 5-slot minimal collection of "knots", where the available "knot" places are very limited, ( so it pays if are not oblized to include the Versatackle or the Trucker s hitch and cover valuable slots with other, more useful knots )(2), the question came back. The hitches, be them TIB or not, are they "knots", or only compound knots, non-local rope mechanisms  ?

1. http://igkt.net/sm/index.php?topic=3610.msg20701#msg20701
2. http://igkt.net/sm/index.php?topic=4418.msg28122#msg28122

[X1]

   Elementary, my dear xarax...    
   After a night s sleep, I awake with a completely different mood - with no more questions, but with some (tentative) answers:
   I thought of the exactly opposite situation : Let us have a proper "knot", a knot s nub, and penetrate it from the one to the other side with a main line or a pole, loaded through both its ends. Will this "knot" become less of a "knot" ? No, it will be simply an ordinary "knot", just wrapped around a "neutral element", regarding its own function. We can now imagine that the cross section of this main line or pole expands, and widens, forcing the "knot' to widen along it ( because it would have to remain tightened around itself, AND around the main line or pole ). So, what will this "knot : become ? A hitch !
   I had already thought of this "thought experiment"  , albeit in a slight.ly different context - but, to my surprize, I had not made the small step needed to actually define what a hitch is, starting from the already conquered ground, the image of the tight, dense, minimizing ropelength tangle we use to call the knot s "nub".
   The interested reader is kindly required to read the following lines, replacing the "neutral element" with "main line" or "pole"
   

    What is most interesting ... is the following related question : Can "neutral", un-loaded rope segments be part of a knot, and prevent this knot of being able to get itself un-knotted, by their mere presence there, by their bulk, the incompressible volume of their rope material ? In other words, can a knot that is interwoven with one or more neutral, not loaded pieces of string, be helped to remain knotted by those pieces ( I suppose that the net [vector] sum of the forces on those string(s), induced by the surrounding knot, would be zero, so that the string(s) can not "pushed" and be thrown out of the knot, by the contact with the the other, loaded parts of the knot. We have to pull one end of a "neutral" string to slip it through the knot, in a similar way we pull a key out of a lock...The pressure induced by the the door on the lock, does not throw the key out !    )
   Inserting a "neutral" piece of rope into a knot, to help this knot remain tied, seems to me an interesting thing we could explore.

   One is not obliged to concentrate about the question of the TIB or not nature of the "surrounding knot", as I had done in this paragraph, which is only of a secondary importance ( a TIB or not TIB hitch, is a hitch nevertheless ). The "surrounding knot" is a not but a knot s nub, a proper "knot", in the case the diameter of the main line or pole shrinks to zero. We are not allowed , of course, to alter the topology of the hitch, that is, no segment of the hitch can cross this axis of the zero diameter cross section of the main line or the pole. Therefore the hitch is bound to be knotted, literally, around this axis, and it is nothing different than a proper "knot", what most people would call the nub of the knot, the local, tight, dense part of it.

[X1]

   ( Second writing )  I decided to keep them both, so there is greater a chance of my awful language could being understood by some brave reader ! 
 
    Elementary, my dear xarax !
    Just after a night s sleep ( and even if I was awaken at one moment by a nightmare involving a certain roodent making grunting noises and trying to bite me    ), I saw the light of the day ! My mood has changed : No more questions, just some (tentative) answers.
    I just made a "thought experiment" of the exact opposite. Instead of starting from a hitch, and trying to see if and how we can define it as a knot, we better start from a "knot", and see if and how we can define it as a hitch. In a split second, everything became much more enlightened !
    Imagine that we have our local, tight, dense tangle, the "nub",  which I have defined as the "knot" proper. How we can make a hitch out of it, literally ?  Simply by penetrating it, from side to side, by a tensioned main line ( in the case of a hitch around a rope ) or by a rigid pole ( in the case of the common rope-to-object hitches ).  What will happen ? The ex-knot will become a hitch, wrapped around the main line ot the pole, and bound to be knotted, literally again, by the mere presence of the main line or the pole inside its core. In short, a "knot" transformed into a hitch, which can not alter its existence, so it was, it is and it will be a knot, ever.
    Now, imagine the diameter of the cross section of this main line or pole becomes zero - and, also, the straight, rigid line of the axis of the main line or the pole becomes curvilinear and flexible. What would have happen ? The ex-hitch would become a knot, again, wrapped around a curvilinear line of zero area cross section. In short, the hitch would become what it never ceased to be : a "knot".
   Why this knot-hitch-knot is a "knot" condemned to remain knotted ? Because of its topology, the fact that no segment of it is allowed to cross this imaginary mail line or pole of zero diameter, so, if it was knotted when it was wrapped around a real main line or pole, it would remain knotted, in theory, and so it should be considered as knotted, when it will be wrapped around this invisible, imaginary path.
    I had discovered that I had already though of something like this, albeit in a slightly different context, which was dealing more with the (secondary) issue of a TIB or not-TIB hitch. However, to my surprize, I had not made the small step needed to be able to define what a hitch is, before examining if there is a difference between TIB and not-TIB hitches. The interested reader is kindly requested to read the following paragraph, thinking of main lines or poles instead of "neutral elements" - and realizing, as I did, today (!) , that what I am actually describing is the relation between the set of knots and its subset, the hitches.

What is most interesting...is the following related question : Can "neutral", un-loaded rope segments be part of a knot, and prevent this knot of being able to get itself un-knotted, by their mere presence there, by their bulk, the incompressible volume of their rope material ? In other words, can a knot that is interwoven with one or more neutral, not loaded pieces of string, be helped to remain knotted by those pieces, but be able to be un-knotted if those stings are pulled off the knot s nub ? ( I suppose that the net [vector] sum of the forces on those string(s), induced by the surrounding knot, would be zero, so that the string(s) can not "pushed" and thrown out of the knot, by the contact with the the other, loaded parts of the knot. We have to pull one end of a "neutral" string to slip it through the knot, in a similar way we pull a key out of a lock...The pressure induced by the the door on the lock, does not throw the key out !   )

 

[SS369]

 

  What is most interesting...is the following related question : Can "neutral", un-loaded rope segments be part of a knot, and prevent this knot of being able to get itself un-knotted, by their mere presence there, by their bulk, the incompressible volume of their rope material ? In other words, can a knot that is interwoven with one or more neutral, not loaded pieces of string, be helped to remain knotted by those pieces, but be able to be un-knotted if those stings are pulled off the knot s nub ? ( I suppose that the net [vector] sum of the forces on those string(s), induced by the surrounding knot, would be zero, so that the string(s) can not "pushed" and thrown out of the knot, by the contact with the the other, loaded parts of the knot. We have to pull one end of a "neutral" string to slip it through the knot, in a similar way we pull a key out of a lock...The pressure induced by the the door on the lock, does not throw the key out !   )

 
[/quote]

The "Neutral" parts are , imo, part of the knot. I don't feel that these so called neutral parts are in fact neutral. They play an important role in those particular knots. Most knots, again my opinion, are compound structures. Yes, we can make a knot that does not do any work, decorative ones for example (though they can do work or a job). A drawn knot or a knot-board's entries does nothing, but they are still knots.

We can see all manner of things within a simple or complex tangle we call a knot. We can even see this in un-knots, such as the Blackwall hitch, that I have used for years as a method of tying off mason's twine to layout foundations of houses.

Hitch: To connect or attach.

So, within many/most knots there are hitches (not all!). A compound structure for sure.

SS

[X1]

 

  [ If the hitch is a knot, then ] Show me your "knot" s nub !

   
 
   ( Third attempt, with an almost one-liner answer ( in comparison to the dull sheets of the previous posts  )

   Imagine that the diameter of the the main line or the pole around which the hitch is wrapped, shrinks to almost zero - but it does not disappear altogether. The wraps of the ex-hitch become rings (toruses) of almost zero diameter, but not balls (spheres). The geometry changes, but the topology does not - because we are not allowed to alter the most fundamental characteristic of any knot ( be it a mathematical, an ideal or a "real", physical knot), its topology. 
   The ex-hitches now becomes  the knot s nub, i.e., a properly defined "knot".  This "knot" remains knotted, even is the hitch was TIB - because no segment of the "knot" is allowed to cross the axis of the main line or the pole. Those elements may not be "functional" parts of the "knot", as the rope segments on which it is tied are, but they still dictate its topology.    
   

[X1]

    Thank you SS369. A brave reader ! 

   The "Neutral" parts are , imo, part of the knot. I don't feel that these so called neutral parts are in fact neutral. They play an important role in those particular knots.

   Sure they do ! However, I had to distinguish them from the segments of the "knot", through which the tensile forces run.
   

   I had to use this simple "thought experiment" in order to clarify what I had meant by this "non-functioning [neutral] elements" neologism - because any segment simply penetrating and/or interwoven within the knot s nub does affect its form, by its mere spatial presence within the knot, by the bulk of its body ( so, it makes a difference, regarding the shape and place of the functioning elements as well ). [The main line or the pole]...do alter the flow of the tensile forces within the functioning elements, although its presence is not necessary for the knot - i.e., if and while the knot is loaded..., it will remain knotted, even in their absence, albeit in a very different form.

[SS369]

    Thank you SS369. A brave reader ! 

   The "Neutral" parts are , imo, part of the knot. I don't feel that these so called neutral parts are in fact neutral. They play an important role in those particular knots.

   Sure they do ! However, I had to distinguish them from the segments of the "knot", through which the tensile forces run.
   

   I had to use this simple "thought experiment" in order to clarify what I had meant by this "non-functioning [neutral] elements" neologism - because any segment simply penetrating and/or interwoven within the knot s nub does affect its form, by its mere spatial presence within the knot, by the bulk of its body ( so, it makes a difference, regarding the shape and place of the functioning elements as well ). [The main line or the pole]...do alter the flow of the tensile forces within the functioning elements, although its presence is not necessary for the knot - i.e., if and while the knot is loaded..., it will remain knotted, even in their absence, albeit in a very different form.

I do not think that we can eliminate the compressive forces and leave just the tensile.

Even though the topology of the segments are the same, with or without the "neutral parts" they both are knots of potential, till they are employed. Or enjoyed.

SS

[X1]

I do not think that we can eliminate the compressive forces and leave just the tensile.

   We can think of "ideal knots", where the circular cross section of the segment is assured - so the compression forces can not and do not alter it. However, I understand what you mean - because you "feel" the compression forces, and the friction, heat, etc, they generate - and there are the compression forces that will lead a "real" knot to its destruction.

[SS369]

I do not think that we can eliminate the compressive forces and leave just the tensile.

   We can think of "ideal knots", where the circular cross section of the segment is assured - so the compression forces can not and do not alter it. However, I understand what you mean - because you "feel" the compression forces, and the friction, heat, etc, they generate - and there are the compression forces that will lead a "real" knot to its destruction.

With "ideal" knots we can "feel" tensile force from one end to the other, but with the addition of compressed point along the route, I don't think there is equal tensile load along the line. Some is absorbed into the body of the tight nub (knot) and transferred.

So for me it is hard to translate the perfect into the real. Like predicting weather or fluid motion.

There are hitches that have nubs as you call it and there are a few that have nothing.

SS

[X1]

There are hitches that have nubs...and there are a few that have nothing.

   If a hitch has one wrap - and all hitches have at least one wrap, of course -, it will have one nub, too - because we are not allowed to alter its topology. Therefore, however small the diameter of the main line or the pole would become, the small ex-wrap = present "ring" would always remain curled, unstraightened, and, together with something else, whatever else would be next to this little curl ( a half hitch, for example ), it will form a "nub". Even a TIB hitch can not disappear, because there would always be one or more little "rings" packed together in a small area, a dense lump of material, a "nub".
   The trick is to imagine the same, topologically, hitch, tied around a diminished, different, geometrically, diameter of a main line or a pole. When you do this, you "see" the nub ! 
    It is the geometry, the large scale of the wraps of the hitch, that does not allow us to "see" the "knot" in it. Change the geometry, shrink the wraps, without altering the topology, and the nub will manifest itself at once. A "knot" can be and can remain knotted due to its topology, and to its topology only. If we do not change its topology, a "knot", any knot, can not change, by definition. So, if a "hitch , when its wraps are shrunk, is a "knot", it should have always been a knot, and should always remain a "knot", whatever the dimensions / the scale of its wraps were or would be.

    Knot tyer, shrink the wraps !   

[SS369]

There are hitches that have nubs...and there are a few that have nothing.

   If a hitch has one wrap - and all hitches have at least one wrap, of course -, it will have one nub, too - because we are not allowed to alter its topology. Therefore, however small the diameter of the main line or the pole would become, the small ex-wrap = present "ring" would always remain curled, unstraightened, and, together with something else, whatever else would be next to this little curl ( a half hitch, for example ), it will form a "nub". Even a TIB hitch can not disappear, because there would always be one or more little "rings" packed together in a small area, a dense lump of material, a "nub".
   The trick is to imagine the same, topologically, hitch, tied around a diminished, different, geometrically, diameter of a main line or a pole. When you do this, you "see" the nub ! 
    It is the geometry, the large scale of the wraps of the hitch, that does not allow us to "see" the "knot" in it. Change the geometry, shrink the wraps, without altering the topology, and the nub will manifest itself at once. A "knot" can be and can remain knotted due to its topology, and to its topology only. If we do not change its topology, a "knot", any knot, can not change, by definition. So, if a "hitch , when its wraps are shrunk, is a "knot", it should have always been a knot, and should always remain a "knot", whatever the dimensions / the scale of its wraps were or would be.

    Knot tyer, shrink the wraps !   

So does this description/definition(?) include such as the Blackwall hitch in your mind? I just see a crossing, no interlacing.
I am fairly sure that I don't consider laying a rope across another constitutes a knot. :-))

Honey, I shrunk the knots..... ;-)))

S

[X1]

So does this description/definition(?) include such as the Blackwall hitch in your mind? I just see a crossing, no interlacing.
I am fairly sure that I don't consider laying a rope across another constitutes a knot. :-))
Honey, I shrunk the knots..... ;-)))

   You got it !   

( You should nt try as hard, and go as far as Blackwall hitch, to discover a counter-example !    ABoK#35, 36, 37 and 49 would be enough ! )
   Yes, indeed, I think that the single ring the Blackwall hitch would be srhunk into, can not pass through the eye of a needle, so it constitutes a nub !
   Any definition reaches its limits, at the limits ! Zero is an even number, but many people would find hard to believe it. Zero to the power of zero is 1 ( 0⁰ = 1 ) - or is it knot ? 
   Also, with the following assumptions:



The following must be true:



Dividing by zero gives :



Simplified, yields :



- Honey, I duplicated everything - including the knots ! 
- Could you, please, honey, duplicate the number of the knot tyers in IGKT Forum, too ?     
 

[SS369]

So does this description/definition(?) include such as the Blackwall hitch in your mind? I just see a crossing, no interlacing.
I am fairly sure that I don't consider laying a rope across another constitutes a knot. :-))
Honey, I shrunk the knots..... ;-)))

   You got it !   

( You should nt try as hard, and go as far as Blackwall hitch, to discover a counter-example !    ABoK#35, 36, 37 and 49 would be enough ! )
   Yes, indeed, I think that the single ring the Blackwall hitch would be srhunk into, can not pass through the eye of a needle, so it constitutes a nub !
   Any definition reaches its limits, at the limits ! Zero is an even number, but many people would find hard to believe it. Zero to the power of zero is 1 ( 0⁰ = 1 ) - or is it knot ? 
   Also, with the following assumptions:



The following must be true:



Dividing by zero gives :



Simplified, yields :



- Honey, I duplicated everything - including the knots ! 
- Could you, please, honey, duplicate the number of the knot tyers in IGKT Forum, too ? 

Wasn't trying too hard to come up with the Blackwall hitch. Just one that I remembered the name of. Can't do much about remembering the ones with numbers as names only.  Point was made and I understand the eye of the needle reference. Still it is hard to accept in real usage.

As far as your math goes, I don't buy into 1=2 unless it is perspective and then anything goes. ;-)))

And I don't buy into that you have duplicated everything. Long way to go!

Duplicating the number of knot tyers in the Forum is an everybody endeavor. Bring someone in...... Another thread please.

SS

[X1]

Can't do much about remembering the ones with numbers as names only. 

   ABoK#35, 36 and 37 are not "hitches" around lines in the ordinary sense, they are the most simple tangles that are topologically equivalent to the unknot, but "knotted" nevertheless, in the most simple way - although their "knottiness" is depending in knothing but geometry, not topology, it is made possible only by the particular location / arrangement of the four ends. However, they, too, pose the same question : Are they "knots" ? And, if they are , where the f... is their nubs ? That is the limit where the "Honey, I shrunk the wraps" picture has been magnified too much, and we can only see some black and white dots, but no shape at all....We can say that it should be expected, and that "any definition reaches its limits, at the limits" - but we can also say that the limits are the ultimate test beds for the validity of all definitions.

[X1]

   The geometry, the geometrical scale of parts of a knot s nub can not change the fact that a knot is a knot is a knot - provided we do not change the topology, of course ! ( If we were allowed to do this, we would have been able to transform every knotted rope into an unknotted one, and vice versa...)
   Change the geometry, shrink the round turns, and you will see the nub that was concealed within them.
 
P.S. The main line or the pole may not be visible - but that does not mean it does not exist !    On the contrary, when we see a "knot" that remains "knotted", although we can not "see" how, we must suppose that it is entangled with some parts of a very small diameter.