Self-tightening knot - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:16:32Z http://mathoverflow.net/feeds/question/85186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85186/self-tightening-knot Self-tightening knot knotted 2012-01-08T13:01:49Z 2012-01-19T20:07:09Z <p>Is there a way, for some finite L>1, to tie two pieces of rope together, such that any finite force is not enough to pull them apart?</p> <p>The type of rope I have in mind is something like cylindrical with radius 1, unbreakable, unstretchable, perfectly flexible, non-self-intersecting and have length L, but I am open to other models. </p> <p>If rope 1 has ends A and B, rope 2 have ends C and D, we tie B and C together, then pull A and D. Is there a knot that holds for every coefficient of friction e>0 and every force F>0 applied to the two ends? </p> http://mathoverflow.net/questions/85186/self-tightening-knot/86135#86135 Answer by Anton Petrunin for Self-tightening knot Anton Petrunin 2012-01-19T20:07:09Z 2012-01-19T20:07:09Z <p>The answer to your question is obviousely "NO", since you want $L$ to be fixed. So let me consider the following question instead:</p> <blockquote> <p>Given the coefficient of friction $e>0$, is there a knot that holds any force? </p> </blockquote> <p>I would bet that the answer to this question is "YES". Obviously we should have $L\to\infty$ as $e\to 0$. </p> <p>A right way to proceed would be to take the Ashley's big book of knots suggested by Matt and look for a knot which admits a sequence of iterations of some kind. Even if you made right guess for iterated knot, actual proof that it holds any force might be difficult.</p> <p>Now let me explain why I would bet for "YES". Assume in addition you are allowed to make any metal ring with zero coefficient of friction with the rope. Then this </p> <p><img src="http://www.math.psu.edu/petrunin/wiki/metal.png" alt="alt text"></p> <p>would solve your problem. The metal ring is marked by black, the green and red ropes alternate; i.e., they go under and over the black ring in turn. (The width of the ring should be about 1.</p>