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BurnOutBrighter6 t1_iqpsh4f wrote

As others have said, yes it's due to entropy. But I want to get to the "why" a little more.

It's probability. There are thousands, millions, maybe infinite different tangled states a cord can be in, and exactly one non tangled state. Being tangled is the "natural state" simply because it's by far the most likely state.
Every time you put the cord in your pocket and scrunch it around, you're picking another random configuration. Since "untangled" is vanishingly unlikely when picking randomly from all possible configurations, it "spontaneously" comes out tangled ~every time.

The exact phenomenon you're looking for is Spontaneous Knotting of an Agitated String by the way. Here's the landmark paper on it:

Spontaneous knotting of an agitated string - Raymer and Smith 2007

Math blog explaining the above paper.

Video of Raymer doing a talk on the topic.

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danby t1_iqtgbe9 wrote

> It's probability. There are thousands, millions, maybe infinite different tangled states a cord can be in, and exactly one non tangled state

This can not possibly be true. Or rather I feel you're conflating topology (does or doesn't contain a knot) with physical pose (the real physical position of a string).

Consider protein folding. No protein forms true knots when they fold but we know there are [nearly] infinitely large number of states any non-trivial protein chain can adopt. And so it must be with regards pieces of string; there are surely an infinite number of unknotted poses a string can adopt and an infinite number of poses which also contain knots.

So yes there is only one topology that has no knots (by definition of the problem) but a string with that topology can still explore nearly infinite numbers of physical states. And this is somewhat reflected in the paper you've linked. In their tumbling experiment their string only knots about 33% of the time. Because there is a very, very large set of unknotted physical states that can be explored.

In the paper they also describe the sequence of braiding steps that generates knots, As this is an ordered set of moves this surely indicates the knotting can't be a purely entropy driven process. Certainly whatever energy surface the string is moving over can't be purely smooth and downhill. Which is likely also why in their tumbling experiment only a third of the tests results in knots, as there is some manner of "energy" barrier that must be surmounted to get to a knotted state.

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