Grip theory
RKT rank¶
As above, the existence of RKT was caused by the existence of a certain stabilizer of two grips. However, not all such stabilizers will enable RKT. On the 3x3x3 grip system, there is a stabilizer of R and L that can only rotate the anti-middle layer (the intersection of the R and L layers) and cannot cause the R layer to emulate another puzzle. In order for the emulation to work, there must be a twist whose axis is R that does not fix L. In general, RKT is possible on a puzzle when there are two grips \(a\) and \(b\) where there is a non-identity element that fixes \(a\) and \(b\), and a different element that fixes \(a\) but moves \(b\).
This can be generalized even further. We can define an RKT rank-0 puzzle to be one with no twists, i.e. one where the stabilizer of any grip is the whole group. Then, we proceed recursively: an RKT rank-\((n+1)\) puzzle is one where there is some grip \(a\) such that the \(a\)-stabilizer of the grip group is an RKT rank-\(n\) puzzle, after removing grips that are stabilized by every element of the stabilizer. This definition correlates with the geometric rank of a puzzle; the RKT rank of a 3n puzzle is \(n-2\) for \(n\geq 2\).
TODO THIS NEEDS DEVELOPMENT