A conjecture about tied knots K, tangled ropes R and tightening forces F.

[xarax]

   Imagine those properties of things :

   Knots K of different geometries.
   Ropes R of different materials.
   Forces F of different sizes.

   Each combination of properties corresponds to one physical object O :
   For example, if knot Kx is tied, rope Ry is tangled, and force Fz is applied on each end, we have the object / materialized knot / knotted material Oxyz = Kx/Ry/Fz - and so on, for all possible combinations of knot geometries, rope materials and force sizes.

   Conjecture about the easiness of untiability :
   If one object O1, tied with knot K1, tangled on rope Ri and tightened by force Fi, is easier to untie than another object O2, tied with knot K2, tangled on the same rope and tightened by the same force, then all objects tied with knot K1, tangled on any rope, and tightened by any force, will be easier to untie than their corresponding objects tied with knot K2.       

[Dan_Lehman]

Or, in other words, only the knot matters!

There is a difficulty here with our needing to define
"knot", and realizing that "geometry" if taken to be
angles & distances and so on will change with force
--and will differ between ropes of different firmness
and elasticity.

Also, ease of untying probably needs to be defined as
some aspect of force required to move a part, because
otherwise a simple refutation is between a complex,
non-jamming knot and a simple jamming one when
given forces that don't jam ... --then the simpler will
be "easier" to untie by virtue of simplicity.
But this argument is really beside the serious point,
just to help shape the need of "force".

I can wonder, though, at "objects" differing in the
slickness of material, where the one of frictive stuff
when set tight is harder to untie, but also resistant
to delivering force well into the knot, whereas the
slick material might do just that and so at higher
forces achieve more tightness!?


--dl*
====

[xarax]

   Or, in other words, only the knot matters !

   I tried to conceal this conclusion under more sentences, as much as I could - and I even added some considerable concessions to your view of the objects we call "knots", as "knotted materials" ( besides my own view, as "materialized knots" ) - but it seems that my diversion did nt work !   

   I can not disagree to anything else you notice - I have only to add that to define=measure the easiness of untiability of a knot is a most difficult thing. An easy or difficult to untie knot, is not a knot which, per se, is more tight and compact, or less tight and compact : It is a knot that WE can untie more easily or less easily. However, there are many "ways" we use to untie a knot : "direct" ways, i.e., by pushing the Standing or the Tail End inwards, and so feed the nub with more material, or "indirect", i.e., by setting parts of the nub in relative motion ( for example, by rotating one part relatively to another ) and so forcing some portion of the Standing or of the Tail End to be swallowed by the nub. Frankly, that was one of the main reasons I had never been able to "test" knots : I had never managed to figure out how on Earth one could "measure" any of those things...