It's too bad that the Wikipedia editors are such jerks; these would be good points to add to the article on Rope.
At 10:16 AM 10/26/2013, you wrote:
if ropes were simply untwisted parallel strands, each a cylinder, and each strand assumed pre-cut at a random location, then pull on rope ==> it comes apart. Well, actually, the different strands will stick due to van der Waals forces, but it'll be very small tensile strength.
Contrast: with suitable twisted construction: apply tension, the outer strands compress the inner ones radially. This causes the strands to stick together due to friction =coeff*(normal force), and we now have positive normal force.
Now note, the normal force actually is *proportional* to applied tension which means no matter how much tension you apply, the rope will not break (assuming unbreakable strands) even though topologically speaking it already is broken by assumption.
So that is a profound theorem. I claim there are related theorems (none proven, but they should be!) about a lot of knots. That is, knots are mechanisms, sometimes very ingenious ones. And for many knots/links, even though topologically the unknot or unlinked, they will not come apart no matter how hard you pull on them (short of breaking) because the design similarly amplifies friction, the harder you pull the more frictional force, so it will not come undone no matter how hard you pull. For example, known rope "bends" (the term of art for tying two ropes together to get an effectively longer rope) will not fail no matter how hard you pull, short of actual breakage. But naive bend designs will fail, like if my mother tied them.
Brilliant knots-as-mechanisms have been designed over the centuries by sailors, truckers, farmers, etc, but mathematicians and physicists never seem to have designed any, nor am I aware of proofs of any of these no-fail theorems.
--Actually, the correct formulation of these "theorems" (for now conjectures) would involve the friction coefficients K for their rope surfaces and the "knot does not fail" claim would only be true if K > threshold (and threshold varies depending on the knot type). (The elastic constants of the rope material and the rope radius both seem irrelevant, kind of cancel out of the question.) Albeit the usual knots found in knot manuals evidently have quite safe thresholds versus realistic K values for actual rope materials. I think there are a few knot types which are not considered safe using slippery m(modern synthetic) rope materials but were considered safe using older, less slippery, rope materials like hemp. The USA actually banned hemp because it is related to marijuana plant and hence plant contains some tiny amount of intoxicating substances, but it used to make excellent ropes in the old days, if it had been banned back then the British empire never would have been. Why do ribbon-shaped ropes (like shoelaces & seat belts) create knots harder to undo than ordinary cylindrical ropes? (If this percepton even is true?) Because have more surface area to be frictionally involved (isoperimetric theorem)? By the way, these thoughts on ropes+knots might be original with me, anyhow they are not discussed in a lot of places you'd think they would be. Which originality is pretty surprising, if true.