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Below is a really hard physics(?) problem which I honestly don't even know how to begin. I thought of it when reading a knot theory book.

Below in the picture, you see the projection of an unknot (a basic looped string in space). enter image description here

The red dot marks an infinite line going into your screen perpendicularly. Assume that we know the line weaves through the "inside" of the loop rather than the outside.

Now imagine the dot starts expanding uniformly as a cylinder outwards. Show that eventually, the unknot shown below will unravel into a circular band that hugs/wraps around the cylinder.

Assume the cylinder has infinite mass and no friction. Assume the unknot is comprised of an idealized string (e.g. no friction, no mass, no volume, infinite ability to twist, no stretching, infinite strength etc.) Assume that there is no other forces that I didn't mention.

Note that I don't even know if this is true, but it seems highly intuitive that when you pull hard enough, any frictionless loop will unravel.

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  • $\begingroup$ Not sure about the knot shown, but this "seems highly intuitive that when you pull hard enough, any frictionless loop will unravel." isn't true for any loop. Take a loop, twist it to make a figure of 8. Bring one side over on top of the other, this makes a double loop, if the cylinder is inside the loop it remains a double loop if it expands., $\endgroup$ Commented Nov 8, 2021 at 23:16
  • $\begingroup$ true, in that case it wouldn't work, but I was thinking specifically of single loops in that it would be highly intuitive. In the single loop case, I couldn't think of what could possibly stop the cylinder from expanding further $\endgroup$
    – kyary
    Commented Nov 8, 2021 at 23:24
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    $\begingroup$ Sorry to be negative, but it seems unlikely to be the case. If the folded figure of 8 (that originated as a single loop) causes a problem, surely there will be lots more loops that wouldn't become single loops as the cylinder expanded. It's a bit of a maths problem by the way, so Maths Stack Exchange might be interested in it. $\endgroup$ Commented Nov 8, 2021 at 23:28
  • $\begingroup$ thanks, I will ask there $\endgroup$
    – kyary
    Commented Nov 8, 2021 at 23:34
  • $\begingroup$ Did you mean that the position of the red dot is always one crossing away from being outside the knot entirely. If so, and if it's an unknot, then your idea may be true, but you should also edit if you meant that, to make it clear. All the best. $\endgroup$ Commented Nov 9, 2021 at 8:32

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I don't know how to solve this, but these two facts might help.

  • Start from the red dot. Move along any path until you reach the outside. The path must cross the rope in doing so. Note that the number of rope crossings is always odd.
  • There is a path to the outside that only crosses one rope.

You probably can use the first point to show the rope winds around the red dot.

The second property probably can be used to show the rope winds around only once.

As the red dot expands, the rope might well form a tangled knot around it. Or it might form a simple loop. I can't tell from inspection.

But you have told me that it is an unknot. If you didn't have the red dot in the way, you could unravel it into a simple loop. I would not be surprised if it is possible with the red dot.

The proof probably involves continuous mappings from one configuration of the rope to another. But I don't know any more than that.

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