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A few weeks ago, my friend radii was puzzling in his server hall. He asked if it was possible to prove that what he wanted to do was impossible, or if he had to remain with his gut feeling. I asked him, and got the following explanation:
I started thinking about the problem, and am now convinced I can prove the impossibility he asked for by basic techniques of knot theory. The argument is what I'll fill this blog post about.
First observation is that due to the size of the fixtures, we can essentially consider the endpoints fixed. Not only that, but since we cannot thread the loops over the fixtures, there's no way to just stick that loose end through any of the loops. So we can basically extract a cube of space and require that all of our modifications be contained completely within this cube, and then stick the endpoints just outside that cube.
This is something called a framed knot, and a popular object of study. My argument is going to manage to steer basically entirely clear of this, though, and I'll just mention it as an extra property to remember.
Equality of knots is basically equivalent to being able to go from the drawing of one knot to the drawing of another knot by the Reidemeister moves. I warmly recommend the wikipedia page for a first glance - these turn out to be absolutely central for constructing knot invariants.
The take home message is that for every crossing configuration, with fixed frame outside, all Reidemeister moves respect the colors at the frame.
Notice that in the diagrams above, the only way to introduce a new color is to already have at least two colors present. Hence, any Reidemeister moves made on this untied configuration will stay entirely green.