Understanding commutator classification

[abunickabhi]

I want to create this post for discussions for two scopes / levels of understanding.

A broad-based understanding of the protocol or methodology to classify commutators for 3D twisty puzzles. Addressing this completely might be a continuous and long endeavour so I do not expect to get all answers here.

The theory part of classification is covered in the ~12+ year old speedsolving wiki article

Commutator - Speedsolving.com Wiki

www.speedsolving.com

We know,
A B A' B' = [A, B]
The commutator has 4 elements, A, A', B, B'.

A basic level of classification can be:

1. Pieces affected by one element (Impossible)
2. Pieces affected by 2 elements
3. Pieces affected by 3 elements
4. Pieces affected by 4 elements (Called as Quirk part - architectural term)

This classified information gives us the different sets of pieces. The pieces not affected by any element is not part of the commutator in any capacity.

The main solvable and definitive outcome I want to get is to classify commutators for blindsolving. There is already enough theory for 3-cycles out there and many people are familiar with list I am pasting below (Reference https://www.speedsolving.com/wiki/index.php?title=BH):

For edges,

1. Half Slice-Planes (4 moves)
2. Slice-Planes (5, 7, 9 moves)
3. Pure Commutators (8 moves)
4. A9/B9 (9 moves)
5. Orthogonals (10 moves)

For corners,

1. Pure Commutators (8 moves)
2. A9 (9 moves)
3. Orthogonals (10 moves)
4. Cyclic Shifts (11 moves)
5. Column Cases (11 moves)
6. Per Specials (12 moves)

Now for 5 cycles or UF5 in my case, these are the type of commutators that I have understood (correct me if I am wrong):

• Double inplace, A and B move pieces inside the intersection
• Overlap with quirk part and without conjugated part
• Overlap without quirk part and with conjugated part
• Overlap with both quirk and conjugated part

Some other classifications taken from the SS wiki article on commutator,

• Single inplace without quirk part,
  • the same transformation is applied twice, the second time in reverse order without interference
  • Examples: corner twist, edge flip, double swap
• Single inplace with quirk part,
  • there is an interference between the conjugated and the non-conjugated version of the transformation
  • Example: [M', U2]