October 28th, 2019, 11:02 AM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory  A knotty outcome in space
Consider a finite string, floating in space, of diameter negligible to its length. Will it eventually become permanently knotted due to randomness? 
October 28th, 2019, 11:13 AM  #2 
Senior Member Joined: Dec 2015 From: Earth Posts: 833 Thanks: 113 Math Focus: Elementary Math 
What I think is no.

October 28th, 2019, 02:50 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,855 Thanks: 744 
What sort of forces are acting on the string? Is the string open or closed?

October 28th, 2019, 03:17 PM  #4 
Senior Member Joined: Jun 2019 From: USA Posts: 386 Thanks: 211  Based on the question, I'm guessing none, other than perhaps small perturbations of insignificant origin? Is that in the spirit of the question? Better question: How thick is the string? If it's molecularly thin (like a polymer chain), tangling seems more likely. If it's multiple strands thick (like thread or yarn), I doubt it will knot itself in the classical sense. (Intuition only, no calculations here.) It takes a significant amount of energy to bend a string away from its neutral position. Don't forget, if we're looking at really long time spans, and the string has some energy, you also have molecular diffusion to consider (much less energy required to dislocate an atom). For that matter, if the string has an energy source (e.g., stellar radiation), it may gas off before it ties itself in a knot. 
October 29th, 2019, 10:48 AM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory 
The string moves as a 2d connected entity, much like the random walk moves as a point. It moves due to zeropoint energy or entropic conditions. The string is open. Its quantity of diameter over length is minimal. (To think of it, I may have seen a problem like this before, i.e., "spontaneous knotting.") 
October 29th, 2019, 11:02 AM  #6 
Senior Member Joined: Dec 2015 From: Earth Posts: 833 Thanks: 113 Math Focus: Elementary Math 
Entropic conditions may lead to chaos .

October 30th, 2019, 01:52 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory  
October 30th, 2019, 02:50 PM  #8 
Senior Member Joined: Jun 2019 From: USA Posts: 386 Thanks: 211  If this is a physical string, made of molecules, then this parameter is not sufficient. We need order of magnitude estimates of the actual thickness, the construction (singlestrand or multistrand, twisted or braided, etc.), and probably some material properties like effective Young's modulus in tension and compression (won't be equal for strings). All of these will affect the amount of energy required for a knot to form ("energy hill/barrier," if you want). I think knowing that is necessary for predicting an expected time scale for knotting to occur, and whether the string is more likely to knot itself or evaporate, for example (or melt, or disappear in proton decay, or whatever).

October 30th, 2019, 03:43 PM  #9  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory  Quote:
I have touched upon most of what you are saying. There must be a Reynoldslike number in it all. Per mathematical theory, does anyone know of a proof that a topological string, with open ends and left to its own designs, will eventually knot? I have read of one.  

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