> 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.