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" ?