Does the amount of friction determine if something is a knot or not ?

[xarax]

   When a "knot" can not remain "knotted", i.e., when it degenerates into a straight segment of rope too easily, even under a minimal pulling of its ends, is it less of a knot ? Is the Grief knot less "knotted" than the Reef knot ? Should some Carrick knots be considered as less "knotted" than some others ? Should the ABoK#1406 and ABoK#1408 be considered as "knots", but not the "similar/identical" ABoK#1407 and ABoK1409 ?
   We know that all practical knots use friction to remain knotted : we do not have practical Gordian knots or Gordian links (1). However, one runs into difficulties when he wants to define "knot-ness" or "knotting-ness" on friction alone. When two ropes are topologically linked to each other ( the simplest case is two linked bights, working like two links of a chain ), intuitively, at least, we tend to consider their configuration as a "knot". ( I am not so sure about the knotted-or-not Gordian links, shown at (1) - but that is not important, because those configurations are not practical knots ).
   So, how much force should a curvilinear segment of rope be able withstand, before it degenerates into a straight line, in order to be called "knotted" ? It is evident that even a Grief knot or an ABoK# 1407 or ABoK#1409 will not become instantly unknotted, when it will be dressed properly/tightly and will be tied on a rough enough rope - but it will slip like the water through our fingers when it is not pre-tightened, or when it is tied on Spectra / Dyneema. I do not believe that "knot-ness" should depend on the particular dressing or the particular friction coefficient of the rope, so a "knot" dressed one way or tied on one material should not considered as "knotted" and remain being a "knot", when dressed differently or tied on another material - but I can not ignore the fact that practical knots are friction machines, and that a machine that is not able to use friction, is no machine at all.
   The same problem remains in the definition of the knot attempted at (2) :
  A knot is any tensioned yet curvilinear segment of a rope, which compresses and is compressed by itself and other segments of ropes.
    If a segment of rope can not remain curvilinear, even under a minimal tension, should it be considered as "knotted" ? Probably not. But then, some "knots" will be considered as "knotted" when tied on ordinary material, and "unknotted", or not-sufficiently-knotted / not-enough-knotted, when tied on Spectra / Dyneema...
    Friction is the measurable property of the flexible materials that makes all practical knots, but it seems that what a "knot" is, and if some segment of rope should be considered "knotted" or not, "for all practical purposes", depends not only on the particular dressing of it, but on the amount of friction present within it as well... 

1.  http://igkt.net/sm/index.php?topic=3610.msg20611#msg20611
2.  http://igkt.net/sm/index.php?topic=4995.msg32926#msg32926

[Dan_Lehman]

   When a "knot" can not remain "knotted",
i.e., when it degenerates into a straight segment of rope too easily,
even under a minimal pulling of its ends,
is it less of a knot ?
(Is the Grief knot less "knotted" than the Reef knot ?)
Should some Carrick knots be considered as less "knotted" than some others ?
Should the ABoK#1406 and ABoK#1408 be considered as "knots",
but not the "similar/identical" ABoK#1407 and ABoK1409 ?

Well, we have at the ready the qualification of
security & stability, if we are generous with "knot".

We know that all practical knots use friction to remain knotted ...

Do we?  At least there are some cases in which
all *ends* of a knot are tensioned and so there can
be no slippage --I write "*ends*" to suggest that indeed
I include the legs of eye knots to be included
(and, e.g., a butterfly knot loaded on its eye and
both tails (or should we say "both S.Parts") will not
slip (given appropriate tensions).

And, so, maybe this is one way to address the issue:
to define "knot" so that it is a structure that endures
full loading (and including an object for hitches & binders);
then, one can use those qualifiers above to speak
of some other *knot* (RFP : choose terms to use!)
that is of non-full loading (which is the more likely case).

But the simple opposition of interlocked bights
is something I think we'd NOT want regarded
as a *knot*; nor a simple turn (or even a full
turn) around a spar.

I believe that my long-ago definition simply answered
the question with a "if, in at least some material, ..."
indication of a knotted tangle being a *knot* --which
leaves a sort of "to-be-continued" appraisal regarding
known materials, but then since the unknown would
be those of some high friction, and we don't really
count Velcro-securing as valid, I think that the definition
is practically workable without surprise (the surprises
coming from new slippery materials that defeat some
*knot* getting its qualification via traditional stuff).

--dl*
====

[xarax]

We know that all practical knots use friction to remain knotted ...

   Do we?  At least there are some cases in which all *ends* of a knot are tensioned and so there can
be no slippage --I write "*ends*" to suggest that indeed I include the legs of eye knots to be included

  In all those cases, the parts of the knot are topologically linked - so, even if there were no friction at all, they would had remained "knotted".
  I have addressed this issue before, in a long but failed discussion with Derek Smith. I am not talking about parts of practical knots which are topologically linked, and so they can not but remain "knotted" - because those parts would had remained "knotted", even in the absence of friction, i.e., in the case of "ideal" ropes and knots. Two linked bights, where all four ends are tensioned, will remain linked, not because of friction, but because of topology - but that is almost a tautology : topology is what can not change by any continuous transformation of the shapes of the objects. I am talking about "knotted" parts which are NOT topologically linked, yet they remain "knotted". In simple, practical knots, there is only one thing that enables them to achieve that : friction. 
   The bend shown in the two attached pictures is, as I believe, a Gordian bend : although the two parts are NOT topologically linked ( as one can see, by examining the topological diagram of the knot : the "red" and the "blue" links can become unlinked without any involvement of their four ends  ), if all ends are tensioned, it will not become unknotted, even in the absence of any friction. It does not require friction at all, its two links remain tangled, because the bulk of the one can not slip through the opening of the other. However, it is not a practical knot, of course. I am not aware of any simpler Gordian bend than this.
   So, I am talking about knots which remain "knotted" although their topology does not "help" them in this - because, when topology "helps", it "dictates", and so there is no point of talking about what would had happened, with more or less friction, as topologically linked "knotted" segments will remain linked, even in the absence of any friction.

[xarax]

define "knot" so that it is a structure that endures full loading (and including an object for hitches & binders);

   A structure in any of its dressings, i.e. a topologically defined structure, regardless its particular geometry, or a geometrically defined structure ? And, in the later case, what are we going to do, when the geometry itself is depending on the amount of loading, so it changes during loading ? Tied in any material , even in the most slippery ones ? "Full loading" until the rope breaks ? "Full" static or dynamic loading ? And what happens when a structure will be able to endure a "full loading" when it is properly dressed and adequately pre-tensioned, but not otherwise ?

   But the simple opposition of interlocked bights is something I think we'd NOT want regarded as a *knot*; nor a simple turn (or even a full turn) around a spar.

   The simple opposition of interlocked bights ( I use a more appropriate term : "interlinked" bights ) is something that is topologically a knot, but nevertheless we can agree not to call it as such, because it does not requires friction to remain "knotted", indeed - and all practical knots use friction.     
   However, what is the difference between a full turn around a spar, and a full turn around a rope, i.e., a nipping loop ? I can not ignore the fact that the nipping loop, the most important element of the king of knots, the bowline, is a knotted part... And I have seen that even when this nipping loop is not "closed", i.e., when it is "opened" and it becomes a helical turn, like it is in the cases of the "Helical loops", it can also play exactly the same role as in the bowline. So, if we will consider the "closed" nipping turn as a "knotted" segment of the rope, we have to do the same for the "open" nipping turns of a Helical loop.

[xarax]

"if, in at least some material, ..." indication of a knotted tangle being a *knot* ...

  Should we include a very soft or elastic, or a very stiff ( for instance, a wire rope ) in those materials ?
  Tangles on very soft or elastic materials, which can be flattened and elongated very easily ( so they do not have a certain, more or less, cross section or rope length ), are able to remain "knotted" because the contact area between their multi-surfaces is maximized, so they differ from knots tied on ordinary ropes quite a lot. Very stiff materials, on the other hand, are able to resist torsion, so the "knots" on them work also very differently from our "practical" knots...

[Dan_Lehman]

define "knot" so that it is a structure that endures full loading (and including an object for hitches & binders);

   A structure in any of its dressings, i.e. a topologically defined structure,
regardless its particular geometry, or a geometrically defined structure ?
And, in the later case, what are we going to do, when the geometry itself is depending
on the amount of loading, so it changes during loading ?

By "full loading", I was addressing the issue of friction
--as you pointed out in citing "topological knots" --:
the point would be to remove un*knotted*/-tangled
apparent tangles : so, pulling all ends and there IS
something tangled no matter, ergo *knot*!
And that leaves much to be desired, then, by some
inner-level discriminations, yes.  (And I did note the
quandry of interlocked bights, which if all ends were
loaded --AHA, in a certain way (such that they were
in fact bights), but not in all ways/directions
--maybe this aspect could free us of such things
(without great gain beyond that, admittedly).)

Tied in any material , even in the most slippery ones ?
...
And what happens when a structure will be able to endure a "full loading"
when it is properly dressed and adequately pre-tensioned, but not otherwise ?

Well, you're confusing things, here : my "full loading"
I explain above --"full" references *ends*, not force
(and thereby shows the move in this formalization
to a not-exactly-at-what-we-call-"knots"-state/entity).

Yes, it is an interesting question about force, though :
one might daily do some things knowing the lightness
of loading --and all works fine-- : are such things to be
discounted?  --maybe akin to finding that the delivery
of forces by HMPE's strong slickness to areas not otherwise
getting them in traditional materials should lead us to
ask if the long-serving structures were *knots*, because
they slip out in HMPE?!

--dl*
====

[xarax]

  By "full loading", I was addressing the issue of friction...
  ...the point would be to remove un*knotted*/-tangled apparent tangles : so, pulling all ends and there IS something tangled no matter, ergo *knot* !

   I see. One can remove "apparent" tangles either by "full loading", or by using an "ideal" rope ( a rope with constant rope length and constant cross section, but no friction ). For all practical purposes, something approaching those two conditions will do the job, I guess : A very heavy loading from all ends and towards all directions of a tangle tied on a very low friction Spectra / Dyneema rope, would reveal if this tangle is a "knot", or not.
   However, those strange beasts, the Gordian knots, will remain "knotted", although they are not topologically linked - so they will slip through this criterion. We can easily release the two parts of the Gordian bend I had shown without any involvement of the four ends, provided we do not pull them all, at all ! 
   The two Gordian links I had shown are very simple, but I can not imagine any "practical" purpose for them - other than a knot-tyers puzzle, perhaps... On the other hand, the Gordian bends may be simple conceptually, but they are obviously quite complicated as tangles - they are very bulky, because they have to be, that is the way they can remain joined, although they are not linked : by pulling all ends, the volume of the one link solidifies, while the opening of the other link narrows, so the former can not slip through the later ( the ropes are supposed to be non-compressible, i.e. retain a constant cross section ). Therefore, I guess we can avoid referring to Gordian knots indeed. when we speak about practical knots, and this removes one, at least, of the many problems we face...