How to find an invariant of the knot puzzle game? Here is the Region Select game based on the knot theory.
Question. Is it possible to find an invariant that divides the initial set of lamps knots into a solvable and an unsolvable game? 
My attempt is:
I have found the papers Dror Bar-Natan, 1992, Ted 
Stanford, 1992. 
 A: If $P$ is a knot projection (rather than a link of multiple components), then any starting configuration of lamps can be turned all off or all on. In Region crossing change is an unknotting operation, Ayaka Shimizu proves that for any fixed lamp, there is a sequence of region flips that can change the given lamp but leave all others fixed. 
The basic idea of the proof follows. Think of the lamps as crossing information, so that the game board is now a knot diagram. Then flipping all the lamps in a region corresponds to changing all crossings around a region. The procedure to change one crossing is:


*

*Orient the knot, and resolve the chosen crossing according to the orientation:

*The resulting link diagram has two components. Apply a checkerboard coloring for one of the components (pretending the other component is not there).

*Perform a region crossing change for each of the shaded regions (in the original knot diagram).


Here is an example of the algorithm in action (taken from an expository article by Shimizu):
Since it's possible to change a single crossing (or flip a single lamp in the original formulation) through a sequence of region flips, that means any configuration of on/off lamps (including all-on and all-off) is obtainable from any initial configuration.
