What is a knot ? Gordian knots.

[xarax]

   A '"knot" is a 3 D configuration - formed  by one, or more, "ropes" (*) - that cannot be transformed into a figure resembling one, or more, separate straight lines and/or perfect circles ( i.e., cannot be "untied") due to one, at least, of the following three conditions :
1.   Topology
2.   Friction
3.   Constant length and cross section.
   Most people believe that, if a knot cannot be untied, this is due to topology or friction only (condition 1 or condition 2). The existence of "Gordian knots" (**), proves that this can also be due to condition 3 only. So, although we  cannot help untie Gordian knots, they, nevertheless, can help us understand and define what the knots that can be untied are.

 (*) A "rope" is a locally elongated, continuous, flexible along its lengthwise dimension 3D object, that :

(1a),
   -In Mathematical knots, studied within "Knot theory", a sub-section of mathematical Topology-
retains its topology (1), but does not have friction (2), or constant length, or constant cross section (3). It is flexible/stretchable in every dimension ; it can even be punched, but it cannot be ruptured.
(1b),
   -in Ideal knots, studied within "theory of Ideal knots", sub sections of mathematical Geometry, and of Computational physics-
does not have friction (2), but retains its topology (1), and has a constant length and cross section (3), i.e. it is flexible only along its lengthwise dimension.

(2),
   -in Physical knots, studied within "Nodeology", by a handful of crazy maniacs in the forum of IGKT   -
does have friction (2), but retains its topology (1), and has (approximately) constant length and cross section (3). ), i.e. it is (substantially) flexible only along its lengthwise dimension/direction.
[ Moreover, physical ropes cannot be bent/make a tight turn around  an object with a diameter smaller than a certain minimum one, and this should perhaps be considered  as a fourth condition - that can prevent a knot tied on physical ropes from being untied. I have not examined this (fourth) condition in this thread.)

 (Note: The attempted description/definition above is about what a knot IS, for itself, and it is not about what a knot DOES, for us, when we use it as a fastening tool. The existence ( structure, form) of a knot is one thing; the application/use of a knot is another.  I have proposed to use this distinction to classify the threads and posts about the knots we are talking about in this forum - instead of a classification based upon the distinction between "theoretical" and "practical  aspects of knots. )  

(**) A "Gordian knot" is an Ideal or a Physical knot that cannot be "untied", even if this is not due to Topology (1), or Friction (2).
    Conjecture:  the only thing that can prevent a Gordian knot to be "untied", other than Topology or Friction, is the fact that the Ideal and the Physical knots have constant length and cross section, i.e. condition (3). That is, the fact that the cross section of the ropes that make ideal and physical knots cannot be altered ( be shrieked or deformed ), together with the fact that they are defined/supposed to be flexible/stretchable only in their transverse dimension, means there might be cases where some part of one knot, that happens to be knotted, due to its volume, its bulk, cannot pass through another part of the same knot that happens to be a small ring. If this is the only way that those two parts can be separated, this knot cannot be untied, i.e. it is a Gordian knot. (At present, I am not aware of any mathematical proof or disproof of this conjecture.)
   Gordian knots have been studied by P. Pieranski et all. (1), (2). They have searched for ideal or physical knots made by one or more separate/not-interlinked closed loops, each topologically equivalent to the unknot, that cannot be untied, even if friction is considered to be zero.
   I present here the simplest two-part Gordian knot/bend I could think of.  ( See the first of the attached pictures). Two topologically unlinked and equivalent to the unknot closed loops/rings are connected in a way that the knot cannot be untied, even if there is no friction present. The only necessary and sufficient condition, is condition 3, i.e. the ropes are considered to be of constant length and cross section . The one closed loop/ring ( the "white") has the minimum constant length required to encircle 4 rope strands at maximum (?each rope strand is supposed to be tangent to two or three others-). I wonder if there can be any simpler Gordian knot/bend than this?
   If we relax the condition that all loops should be equivalent to the unknot, the simplest link/bend I can think of is shown in the second of the attached pictures.  The "orange-red" link is a closed loop with two overhand knots tied into it. The "white link" has the minimum constant length required to encircle 2 rope strands, it is a closed loop able to encircle 2-rope-strands-at-maximum. If we use a 4-ropes-strands-at-maximum "white" link, we can connect together two or more links like the "orange-red" one, and form a "Gordian chain", as long as we wish it to be.  

1)   http://arxiv.org/abs/physics/0103080
2)   http://www.maa.org/devlin/devlin_9_01.html
3)   http://www.guardian.co.uk/science/2001/sep/13/physicalsciences.highereducation

[xarax]

   Well, "evidently" true things can not always be proved !   That happens to be the case for those two simple Gordian bends/links presented in this thread. I have just been kindly informed by an expert mathematician on the field of links, Rob Kusner, ( Dept. of Mathematics & Statistics, at the University of Massachusetts, Amherst, Massachusetts, USA ),  that those links
  '...both illustrate we still don't have many mathematical tools: neither to *prove* the orange "unlink" (at the top), or the "connect-sum" of trefoils (at the bottom), cannot be slipped through the white ring, nor to derive sharp bounds on the lengths of the shortest white rings that *can* be slipped off!!"
   I suspect that, with a multi-folded orange link, that pass through a sufficiently narrow white link more times, our situation does not improve !
   What interests us here is that knots do not need to "use" their topology or friction to remain untied. The mere existence of the volume of a knotted segment of a rope, due to the bulk of the ropes themselves, can prevent it from slipping through a sufficiently narrow loop - even if there is minimal friction involved, or no friction at all. In such a case, a knotted part of a knot should perhaps be considered as a stopper sub-knot into the knot, a stopper that can not move freely inside the rest of the knot s nub.
 

[Dan_Lehman]

Ah, you dare to step into blue water !

This sounds, but for diction, similar to my musings.  I thought
of "linear" as appropriate --a linear structure.  (Actually, I used
"curvilinear", but have since wondered if this goes further than
needed --i.e., that the "curv..." aspect will follow necessarily
from the entanglement?  Perhaps if not restrictive, the descriptive
aspect reinforces the notion, over mere "linear"?)

I adopted "PoFM" (Piece of Flexible Material) and did specify
"uniform cross-section".  I want to speak of "knottable media"
and to include our common flat media --webbing/tape (both
tubular & solid (not "flat", please) --note that the former allows
of some entanglements impossible in rope (i.e., insertion)).

Which is not to say that we might not treat such things
as chain, but must regard them as exceptional/abnormal.

You try to limit the entities *negatively* --by stating what
they canNOT do ("... into a straight line").  I tried to think
of some *positive* conditions --be a "structure" that endures
tension.  Yours might be a cleaner path to this desired condition.

But I think that the foundation should be like mathematician
John Conway uses, of his "tangles" : that one sees END-2-END
segments of PoFM entangled.  This precludes there being any
pure circle.

In order to make your requirement of "cannot be transformed ..."
valid, it must be the case that there is some "cookie cutter"
boundary put atop the *knot* such that all *ends* ("limbs"?)
run beyond this boundary into non-considered space --i.e.,
the *knot* is within, entirely.  (Otherwise, any TIB knot,
e.g., can be transformed into a non-knot.) [Tied In Bight]

Which puts the rub on Chisholm's "nub", which he saw as
the *knotted* PART of e.g. an eyeknot, with the eye being
part of the *knot* beyond this "nub".
For the purposes of getting a good, sharp conceptual model,
it might be necessary to abandon this way of dealing with
such aspects, and to treat them somehow otherwise.

I don't see how your conception treats hitches & binders
--i.e., entanglements that involve non-PoFM parts?
(And I regard hitching or binding other PoFM as often
in this condition.)  A mere turn around a pole cannot
"be transformed ... ", but is that a *knot*?  Or am I just
missing that the mere un-turning/-wrapping is valid as
a transformation; which would not alter a clove hitch
or constrictor unless one untied it.

--dl*
====

[struktor]

Numerical simulation solution do not Gordian knots:
http://etacar.put.poznan.pl/piotr.pieranski/Non-GordianHalfSize.gif


http://etacar.put.poznan.pl/piotr.pieranski/GordianUnknots.html
The simplest problem one can pose here is: are there such a entangled conformations of the unknot tied on a finite piece of rope,
which cannot be disentangled to the ground state torus conformation?

This is a non-trivial problem. A firm answer to the question is not easy to find.
As the numerical simulations we performed indicate, the answer is positive:
there exist conformations of a the unknot, entangled in such a manner, that they cannot be disentangled to the torus conformation.

Struktor

[xarax]

the "curv..." aspect will follow necessarily from the entanglement? 

   To be entangled, a piece of material should be curvilinear or piecewise linear, i.e. made of linear segments connected together end-to-end. If we do not consider the "fourth" condition mentioned - that the radius of a physical rope s curvature can not be smaller than a minimal number (*)- the "curvilinear" characteristic will necessarily follow from the requirement of "entanglement" (**), I believe. However, the term "curvilinear" ,as an adjective, reinforces the intuitive picture of the entangled piece(s) of rope, and this is good for our purposes.   
  (*) In 3D space, we have two "principal" curvatures, and two radiuses, as the surface of the rope is curved along two dimensions.
(***) "Entanglement" itself is not a notion that is so evident in my mind...If two loops  can be continuously transformed, so that, in some configuration, their spatial separation is bigger than half their rope length, that are not entangled, and this proves that they were never entangled. It seems to me that this operation is a sufficient, simple, intuitive way to define not-entangled loops, but I do not know if it completely satisfactory for our purposes.
   If we wish to "see" if two loops are entangled or not, without having to envision any 3D transformation of themselves in the configuration we are examining, we can try a different picture : Consider each one of those loops as the boundary of a circular disc - however deformed this disc might be.  If those two discs can not but intersect each other, the two loops are entangled. This way we define "entanglement" with the help of "intersection", which is a more simple, intuitive notion.

"uniform cross-section".

  If "uniform" means also "of the same shape and area"(*), It might be a better term, indeed, because it "shows" explicitly that any cross section of the rope remains the same alongside its lengthwise dimension.( I did not used this term at the first place, because I though that the "form" of  a thing, might mean only its shape, irrespectively of the scale of this shape. ) I feel that the term "uniform" refers to something that can be measured in many positions, while "constant" describes only the final result of the comparison of any such measurements. I do not know the language sufficiently well, to understand those subtle, for me, semantic nuances of those words...
(*) If the shape remains the same, but the area is allowed to diminish as much as we wish, condition (3) does not apply, of course, and the two examples I give in this thread can be untied.

... the former [tubular/hollow flexible materials] allows of some entanglements impossible in rope (i.e., insertion)).
...must regard them as exceptional/abnormal.

THAT is an understatement !  It had not crossed my mind...A knot made by the passage of a segment of a rope through the tubular/hollow cross section of the segment of the same(??) or another rope...Hmmm...

I tried to think of some *positive* conditions --be a "structure" that endures tension.

   It would be nice to be able to do this, but I had to include Ideal knots into my definition- where we do not have any friction. Indeed, what is the surprising thing about Gordian knots and links, is that they do not need the "use" of neither their topology, nor their friction to achieve un-knotting-ness. To me, that reinforced the appreciation of the fact - seen in some practical knots- that there might be parts of a knot which serves in its working, without themselves be tensioned, i.e. , by their mere bulk, the volume of the rope material they contain. ( Rope is flexible, but not compressible). In other words, those parts may be considered as stoppers within the knot they belong, that prevent the slippage of the tail by the same Gordian knot/link mechanism : A knotted sub-knot can not pass through a sufficiently narrow loop.

Otherwise, any TIB knot,...can be transformed into a non-knot.

   As regards Ideal knots, that is the case indeed. I have tried o include Mathematical AND Ideal knots into my definition of knots, and knot s un-knottability conditions.
To include the TIB knots, we have just to connect their free ends, and examine the situation beyond this stage. Only loops can be parts of Gordian-like structures.
   What is most interesting, and, at present, completely unknown - to me, at least - 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 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 !    )   
 

I don't see how your conception treats hitches & binders--i.e., entanglements that involve non-PoFM parts? (And I regard hitching or binding other PoFM as often
in this condition.)

   The flexibility or not of the pole is irrelevant. As I have said earlier, we have to think of those elements (poles, rods) as loops, (as rings), i.e. imagine that their ends are connected, before we examine if the knot can be un-knotted or not. A mere turn around a pole cannot " be transformed ... ", indeed, but it is only a *knot* that has to "use" its topology to remain un-knotted. (condition 1) ( The pole should be regarded as one ring, and the loop as another ring, topologically connected to the first one ). Do not forget, we wished to define all knots, that is also knots  that remain un-knotted irrespectively of topology  or friction ! ( Do not have to "use" topology and/or friction). Then we will proceed to knots that have to use friction, as our practical knots - with open ends - are.  If we achieve this first difficult task, all remaining things will be a piece of cake, I believe.

PS. Where this ingenious, prolific mathematician (John Conway) had talked (ALSO!) about the definition of the "entanglement" ?

[xarax]

   Thank you, struktor,

are there such a entangled conformations of the unknot tied on a finite piece of rope,
which cannot be disentangled to the ground state torus conformation?

  Yes, indeed, that is the special problem of the single-loop Gordian knots. In this thread, I have only presented two two-loops Gordian knots/links , i.e. two parts bends.
  For a possible single-loop Gordian knot, see the attached pictures. Notice that only half of the knot, the one of the two symmetric parts, is shown.  ( Cut and paste the picture, to see the whole knot). The solution I have though of, is, essentially, a pile hitch over a sufficiently knotted double line end of the loop, so that this knotted part is large to prevent the pile hitch go over and out of it. If the length of the rope is sufficiently shrunken, we have a simple (conceptually !) Gordian knot on a single loop,  I believe. Again, it is a part of the knot that is itself knotted, which plays the role of a stopper into the knot s nub. This part, due to its bulk. the incompressible volume of its rope material, can not slip through the narrow bight of the pile hitch, so the knot can not be untied.
  

[SS369]

I just feel compelled to inject that though this study of knots, mathematical, etc. will help us understand the working directions the flexible material can take, I feel it is limited till such time as we can add in the compressible properties of the flexible linear material.
That is the proverbial fly in the ointment.
Can this mathematical study help us to design or prove a bend, loop, hitch, etc. that with no aspect of friction and compression, will work? Work without let's say a stopper of some sort?

SS

[xarax]

   Thank you SS369,
   The compressible properties of the ropes could also be taken into account, and yet those Gordian knots/links remain Gordian - as long as there is a non-zero limit in their possible compression, and the maximum allowed dimensions - of the link through which we have to pass the knotted part(s) of the rope(s) -, are suitably decreased. Obviously, it is the fact that the cross section of a physical rope can not be shrunken to a point, what prevents knotted section(s) of this rope from been able to pass through a sufficiently narrow link.
   Now, just notice that even the most simple examples I have presented are not mathematically proved, to the time being, even under those more general, and more easy to be taken into account, compression-free conditions. Let us make some first steps into those simplified, idealized situations, and then we will proceed to more physically accurate models. I can assure you that the assumption of the zero, or limited, compression of the rope material, is the last thing we have to worry about !  For example, I guess that the condition 4, not examined in this thread, should be the sole reason that some knots would not be able to be untied, although I was not able to figure out simple telling examples, as those given for the condition 3.

[Dan_Lehman]

Are these Gordian knots endless?  --i.e., there is an unknotted
ring then knotted and shrunk to this ideal snugness that then
doesn't allow untying.  But with ends, it would take friction to
keep them from being able to be backed out, in theory.

--dl*
====

[xarax]

Are these Gordian knots endless?  ... with ends, it would take friction to
keep them from being able to be backed out, in theory.

   Yes, only "endless" knotted  unknotted ropes , i.e. loops, can be Gordian. But that does not prove that there would nessesarily exist any Gordian knots with one or more loops, that does not depend upon topology or friction to remain unknotted ! We have to discover some such knots, and we have to prove they are Gordian indeed, and this has be done till now only by the means of computational tools, and for only a few knots. As we speak, there is at least one mathematician that tries to prove that the first Gordian link I have presented is indeed a Gordian link, and calculate the maximum ropelength the "white" link should have, in order to prevent the "orange" strands and their bights to slip through it. Not an easy task !  
   I take the liberty to repeat here something that has to do with segments of loops, i.e. pieces of ropes with ends.
     What is most interesting, and, at present, completely unknown - to me, at least - 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 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 !    )
   Insert a "neutral" piece of rope into a knot, to help this knot remain tied, seems to me an interesting thing we could explore.

  

[Dan_Lehman]

Can this mathematical study help us to design or prove a bend, loop, hitch, etc. that with no aspect of friction and compression, will work?

I'm not interested in mathematical knots,
only physical ones.  Such that mathematical insights
can help, fine.  I don't see the Gordian-knot exploration
as terribly helpful.  Meanwhile, it is yet a struggle to shape
a concept of *knot*!  --though two of us seem to have
leaned in a similar direction, and that is a happy circumstance.

--dl*
====

[xarax]

   Unfortunatelly, I am not able to be interested in mathematical knots either. However, I believe that, if we wish to understand what a physical knot really is, we have to start from where mathematical knots have already arrived. I was surprised by the fact that friction is not the only thing that enables one or more tangled pieces of rope to remain knotted... I was also surprised by the simplicity of the two Gordian links I have presented in this thread, and by the fact that, for the time being, there is not any mathematical proof of their "evident" untieability. Also, studying Gordian structures, I believe I have understood better the role which the bulk of a knotted segment of the rope that makes a knot can play  - even if the rope in this segment is not loaded, or it is evenly loaded from both sizes. ( I have even figured out a TIB end-of-line fixed loop that utilizes this "principle" : a bight goes through the ring, and encircles a sufficienty voluminous knotted structure at the standing part, in a way that the rim of the bight can not slip over it, and enable the loop to be released...See attached pictures)
   Until now, I believed that the bulk of an area of a knot, ( which we can easily increase, by retuckings, for example, or by retraced or double line paths ), can only serve as a means to obtain greater first curves, by forcing the standing part to encircle this area - and as a "cushion", able to absorb dynamic loadings. Now, I believe I can also "see" better how the mere volume of a section of a knot is an obstacle that can prevent this part of the knot move/slip through the same, or through another knot. This function of volume in knotting may sound obvious to many people, but I had not realised it so clearly as I think I have now.

[xarax]

   Two Gordian bight-2-bight bends (See 1), that do not use topology or friction.
   The mechanism is the same as the one presented at Reply#5 :
A Pile hitch over and around a voluminous knotted segment of the one bight end, interlinked with the other bight end.

1)  http://igkt.net/sm/index.php?topic=3640

[struktor]

Bayman method takes into account the impact of friction.
http://www.cs.cmu.edu/afs/cs/academic/class/16741-s07/www/projects06/stolarz.pdf

[DerekSmith]

[ Moreover, physical ropes cannot be bent/make a tight turn around  an object with a diameter smaller than a certain minimum one, and this should perhaps be considered  as a fourth condition - that can prevent a knot tied on physical ropes from being untied. I have not examined this (fourth) condition in this thread.)

Although this constraint has not been elaborated upon, I would like to take exception to it.

Of the cordage I have thus far worked with, I have yet to come across one that cannot be turned around a diameter of zero.  I don't doubt that there may be exceptions, but to my experience, the claimed constraint does not hold for the majority of cordages.

Derek