We now adapt this game to a more general and more abstract setting where the physical constraints of the problem are gone. Suppose we represent the initial knot with a 2-regular graph inscribed in a circle (i.e., we have a graph with n vertices with exactly two edges incident on each vertex). Initially, some edges may cross other edges and this is undesirable. This is the "knot" we wish to unwind.
A "move" involves moving any vertex to a new position on the circle, keeping its edges intact. Our goal is to make the fewest possible moves such that we obtain one n-sided polygon with no edge-crossings remaining.
For example, here is a knot on 4 vertices inscribed in a circle, but two edges cross each other. By moving vertex 4 down to the position between 2 and 3, a graph without edge-crossings emerges. This was achieved in a single move, which is clearly optimal in this case.
When n is larger, things may not be quite as clear. Below we see a knot on 6 vertices. We might consider moving vertex 4 between 5 and 6, then vertex 5 between 1 and 2, and finally vertex 6 between 3 and 4; this unwinds the knot in 3 moves.
But clearly we can unwind the same knot in only two moves:
6 4 5 3 5 2 6 1 6 1 2 3 4 6 2 6 1 4 5 6 2 5 3 4 1 3 0
Knot solvable. 2 Knot solvable. 1