Author Topic: What is a knot ? Gordian knots.  (Read 23161 times)

SS369

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Re: What is a knot ? Gordian knots.
« Reply #30 on: December 27, 2011, 05:24:17 PM »
Hi xarax.

I think that the bundles of fibers, whether braided or twisted, do slide. But that is most likely a very small slide, more like a small movement. The reason for the wrap of "plastic" around the core strands in one of your ropes is most like to protect them from each other since some kern fibers that are very strong are also frictive/abrasive to each other. The inner sheath holds the bundle relative and the outer sheath is the protective mantle.

I personally think that the core issue here is more of the tensile and compressive nature of the relationship of the components. Some hard laid strands just don't have the same amount of room between the fibers and will only compress and elongate so far and have counter rotated strands next to them, while braids have more space within to do more moving.

So as for sliding, I think they do just not very much.

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xarax

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Re: What is a knot ? Gordian knots.
« Reply #31 on: December 27, 2011, 06:49:59 PM »
The reason for the wrap of "plastic" around the core strands in one of your ropes is most like to protect them from each other, since some kern fibers that are very strong, are also frictive/abrasive to each other.

  Believe it or not, I had not thought of this simple and obvious to me (now) explanation !  :) Thank you. So, this spiral tube made by this plastic tape, is just  protecting the fibers from abrasive contact, and it is not playing any other role. 

... hard laid strands just don't have the same amount of room between the fibers... while braids have more space within to do more moving.

   So, you say that a bundle of many sub-bundles with braided fibers, is more fxexible than a bundle with the same number of sub-buntles, but with twisted, laid-woven fibers ? The black-yellow rope shown, although it does not have any laid woven elements, but only this tightly woven inner sheath, is VERY stiff... Probably because the fibers of each sub-buntle, being parallel  to each other, do not have any  room to move, and the tightly woven inner sheath keeps them even more close to each other.  I am guessing that parallel is tighter to laid , as laid is tighter to braided, and that, if you put parallel fibers into a tightly woven braided inner sheat, you have a very stiff rope.
« Last Edit: December 27, 2011, 06:50:34 PM by xarax »
This is not a knot.

SS369

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Re: What is a knot ? Gordian knots.
« Reply #32 on: December 27, 2011, 07:33:17 PM »
The reason for the wrap of "plastic" around the core strands in one of your ropes is most like to protect them from each other, since some kern fibers that are very strong, are also frictive/abrasive to each other.

  Believe it or not, I had not thought of this simple and obvious to me (now) explanation !  :) Thank you. So, this spiral tube made by this plastic tape, is just  protecting the fibers from abrasive contact, and it is not playing any other role. 

... hard laid strands just don't have the same amount of room between the fibers... while braids have more space within to do more moving.

   So, you say that a bundle of many sub-bundles with braided fibers, is more fxexible than a bundle with the same number of sub-buntles, but with twisted, laid-woven fibers ? The black-yellow rope shown, although it does not have any laid woven elements, but only this tightly woven inner sheath, is VERY stiff... Probably because the fibers of each sub-buntle, being parallel  to each other, do not have any  room to move, and the tightly woven inner sheath keeps them even more close to each other.  I am guessing that parallel is tighter to laid , as laid is tighter to braided, and that, if you put parallel fibers into a tightly woven braided inner sheat, you have a very stiff rope.
The plastic core wrapper(s) may have, additional services, e.g., water resistance, methods of actual manufacturing, etc. That may be a stretch, but I think it is primarily to insulate the fiber bundles.

Yes it is my understanding that a core bundle of braided material will be more flexible because of the many directions the fibers are woven than a hard laid (twisted) core bundle of "parallel" fibers.

The closer to perpendicular, to the path of the rope's direction, the spiral is the more spring there will be in the resistance to lengthening. The straighter the fibers are, the more stiff in tension it will be. A rope of perfectly straight fibers, although "strongest" will be the least forgiving if shock absorption is a criteria.
The stiffness to knotting and bending will be a direct correlation to the density of the bundle.

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xarax

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Re: What is a knot ? Gordian knots.
« Reply #33 on: December 27, 2011, 10:11:12 PM »
   I have tried to imagine what will happen, if those individual bundles of parallel fibers are only loosely  included into a flexible outer envelope/sheath, so they can slide  and reposition  themselves inside the it (1). I guessed that a rope made this way will be a knot-friendly rope, because the knots tied on it would be more like multi-line knots - and multi line knots are stronger than single line ropes (of the same total cross section area).
   Let s say we wrap loosely 6 such bundles into a compound rope like this. When bent around a tight curve, it is expected that the cross section of such a rope will be flattened easily and maximally, and there will be three bundles running the inside, shortest path, and three bundles running the outside, longest path  (i.e., the cross section would become rectangular, like that of a strap ). The length difference of the two sets of paths around the curve will be smaller than those of a normal rope, where the cross section have to remain more or less circular. Moreover, the individual bundles, acting like individual lines, will adjust themselves inside the knot s nub as well, and they will be able to absorb the tensile forces collectively, so more effectively : there will be more than one "weak links" in sections maximally loaded and/or curved, so I guess that such a multi-line rope will break later than a normal rope. We would have a rope that can bend around curves of smaller diameter, and a rope that can deform itself, so it can absorb maximum loading into the knot s nub more effectively. To remain round when not maximally loaded, such a rope can have its individual bundles arranged around an soft, elastic central core, which could be easily compressed and deform, to allow the stiff individual bundles of parallel fibers to reposition themselves when the rope is bent.
   
1. http://igkt.net/sm/index.php?topic=3169
This is not a knot.

SS369

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Re: What is a knot ? Gordian knots.
« Reply #34 on: December 28, 2011, 12:43:32 AM »
One of the things that immediately comes to mind with this able-to-flatten rope is that there will be bunching of the outer sheath along with the tendency of the rope to "milk". By that I mean that the core and sheath will move too independently.

Conceivably there could be parts within the knot where the mantle is wrinkled causing the core fibers to have higher loads induced against them.

It is on the recommendation of industrial sling manufacturers that no knots be employed at all, ever.  Though they do not discourage the use of a a choker hitch set up. I have two endless round slings that are core strands within a loose fitting sheath.

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xarax

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Re: What is a knot ? Gordian knots.
« Reply #35 on: December 28, 2011, 03:03:44 AM »
...able-to-flatten rope
... the core and sheath will move too independently.
...there could be parts within the knot where the mantle is wrinkled

   I like this able-to-flatten adjective ! :) That is the idea, to have many almost independent, sliding cores, within the same loose encompassing sheath ( that should be very flexible, perhaps also elastic...). As the mantle is always secondary, I do not think that any wrinkles would cause much trumble in the tension bearing capabilities of such a compound, many-lines rope. Yet another thing one has to test, I guess...

This is not a knot.

X1

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Re: What is a knot ? Gordian knots.
« Reply #36 on: June 29, 2012, 05:13:33 PM »
A KnotTyer (1) diagram of the Gordian bend, presented at Re#12 :
http://igkt.net/sm/index.php?topic=3610.msg20970#msg20970
   Here is an attempt to use this bend as a counter-example against the (false) claim that : " Topology determines Geometry" :
   Let us suppose we do not have access to any of the four ends of a bend. We can loosen this rope tangle, re-arrange it, and then tighten it again. Will we get the same knot ?   Nooo! We may well get another knot, topologically equivalent but geometrically quite different. So, topologically identicall tangles, when they are dressed differently, generate geometrically different knots. Moreover, there can be cases where the initial tangle, when tightened, can lead to a knot, or not. The Gordian bend shown here has the same topology with two not-linked open loops. We can loosen it, without changing the topology of its links, and tighten it again. Depending upon the particular dressing, at the end of the tightening we may get a secure tight bend, or two loose unknotted loops ! The interest thing is that this fact is not depending upon the physical characteristics the ropes. Any  bights with inaccessible ends, however slippery the ropes they are made of would be, can be linked by this bend.

in other words;

" At the attached files, you can see a 2D diagram and pictures of a bend I call " Gordian" . It is a link between two closed bights/loops, where we do not have access to their one tip. ( I have drawn these inaccessible "ends" as been warped around two poles, to show that we cannot use them, but also for decorative purposes...)
The most interesting/important thing is that this bend does not use friction to work ! It will work even if the ropes well 'ideal" - in the same sense we call the abstract mathematical structures that some mathematicians and physicists study, as " ideal knots" . The two links will not slip the one through other, because in their way to do this, they encounter obstacles imposed by the volume, the bulk of knots tied on the ropes. In this particular bend, there is a double line overhand knot tied on the one link (blue/white), and the accessible bight of this link is warped around it and it cannot overcome it. The bight of the other link (red) is trapped in this tangle.

Of course, we can untangle the bend, in the same way we have tied it in the first place. However, once we have tightened the bend, to be able to untie it this way, we have first to set the bend a little bit loose, to un-tighten it for a while. Then we can pass the bight over the bulk of the double line overhand knot, pass it through out the other s link bight, and release the tangle of the two closed loops.
So, once the bend is started to be tightened, and this tightening does not stop and is not reversed for a while, the bend can not be untied in an way - unless you cut the rope ! ( This is the reason I have called it " Gordian bend' ) Also, do not forget, this is independent of the low or high friction of the ropes, it will have happened even if our ropes were infinitely slippery, i.e. if they were "ideal' ropes.

The 2D diagram of this bend is the one shown. at the attached file. However, the topologically equivalent 2D diagram, is just two not-linked closed loops ! You can pass from the one diagram to the other, without having access to the ends of the ropes.

So, topology DOES NOT always determine the geometry of the knots, even if these knots are "ideal" knots."

1) http://daveroot.co.cc/KnotMaker/
« Last Edit: June 29, 2012, 05:17:39 PM by X1 »