"impossible" ring puzzle in Japan: would a topologist figure it out in no time?

Factual Questions
21

You called? I am a homotopy theorist, though not a knot theorist. [Sorry, reminds me of a visiting lecturer from England when I was a grad student, who would refer to spaces confusingly as “not nought connected.”] If I found the right video (the original URL didn’t work for me), this is not so much a knot but a link (involving the rope and the wooden ring). Or a knot embedded in a half space from which has been removed a circle (the wooden ring) and a line (the wooden pole). And a nontrivial link/knot at that, or you would be able to just take the puzzle apart.

Topologically, if you were allowed to shrink the size of the ring, it wouldn’t matter what the link was, you could always slide the ring along the rope. So it’s not purely a topological problem: The practical problem is whether there is a series of steps that exposes all parts of the rope in succession to one side of the fixed wooden ring as the metal ring slides along.

The first step of the solution in the video converts the link (by a homotopy!) into a form where it’s pretty obvious what the solution will be from that point on.

So, if we were to use link-theoretic tools, I suppose it would be to use some invariant that shows that that the original link can be deformed into the modified link in that first step. But I’m not immediately seeing any invariants that would address the real practical problem that needs to be solved to solve this puzzle.

22

Right, I recognized this, but presumed one could formalize the constraint of the metal ring not passing through the wooden ring simply enough, without having to treat it as a geometric (i.e., size) problem at that stage, but perhaps formalizing that is not so simple. Anyway, thanks for providing a topologists’s perspective!

23

I can’t immediately think of a good way to account for that constraint topologically, but if anything occurs to me, I’ll let you know.

24

OK, I’ve given it some more thought. I’ll sketch what I think is going on, but I haven’t worked out a formal proof.

First, knots and links are red herrings. That’s not what’s really going on here. It’s really a homotopy theory issue. The following is going to address only the existence of a solution, but buried in the argument I think is also a way to find a solution.

Start at one end of the rope, follow it to the other end, and watch how the rope passes through the wooden ring. Straighten it out a bit so that, each time it passes through, it passes straight through. Count, say, a passage from the front of the ring to the back as +1 and a passage from back to front as -1. Add those up and call that the “crossing number.” (The mathematicians reading will recognize this as a version of the winding number or as an oblique reference to the fundamental group of the complement of the ring-and-pole.) No matter how you manipulate the rope, this crossing number will remain invariant. And the important point is that it is, in fact, 0 for this puzzle.

We could consider variants of this puzzle with the rope arranged differently. My claim is that any such puzzle is solvable if and only if the crossing number is 0 (assuming a sufficiently long or elastic rope).

The metal ring, not being able to pass through the wooden ring, can never follow a path with nonzero crossing number. That gives you the only if: If the puzzle is solvable, the crossing number must be 0.

Less trivial is the claim that, if the crossing number is 0, then the puzzle is solvable. Here’s where I get (more) sketchy. A consequence of the crossing number being 0 is that, if we allowed the rope to pass through itself, we could remove it from the wooden ring entirely. Since it can’t, actually, cross through itself, the best we can do is pull whatever part of the rope we need through the wooden ring to the correct side, with some other part getting mildly in the way. But, given a sufficient length of rope, not so badly in the way that we can’t pass the metal ring along the part of the rope we need to. So the puzzle is solvable.

For the mathematicians: The way I think you can make this rigorous is to consider the universal covering space of the complement of the ring and pole (which I picture, schematically, as an infinitely long tube), and the infinitely many lifts of the rope to this space. Any one of these lifts will start and end in the same region of the tube, because the crossing number is 0. But the lifts may be interleaved in such a way that you can’t simultaneously move them all into their own separate regions (= you can’t remove the rope from the wooden ring without passing it through itself). As we manipulate the rope in our world, each of its lifts moves in the covering space, maintaining the interleaving. Consider the metal ring to be on one of these lifts, but restricted in its movements to just a finite region of the “tube” (= it can’t pass through the wooden ring). We can move the ring along its rope if we pull that part of the rope we want to move along into the finite region, at the expense of moving another part of the rope (i.e., part of the next lift that’s interleaved with our copy) out of the region. But that’s fine. At the end we’ve moved the ring from one end of the rope to the other.