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.