Commutator

A commutator is a sequence of moves that consists in doing a sequence A, then a sequence B, then the inverse of the sequence A and finally the inverse of the sequence B. As a result, only specific pieces are affected. It allows to finish the cube without disturbing pieces that are already in place.

Mathematical definition

Given a group, a commutator is an element of the form ghg'h' (also denoted [g, h]), where g and h are elements of the group with inverses g' and h'.

Cube notation is very close : A B A' B' = [A, B]

Fundamental principle

It is straightforward that A A' does nothing, as well as B B', because A' is the inverse of A and B' the inverse of B.

So the commutator A B A' B' does change something only because B A' is not the same as A' B. When B A' = A' B, or B A = A B, we say that the elements A and B commute. In this case, A B A' B' = A (B A') B' = A (A' B) B' = I where I is the identity.

For example for L and R or U and D. In this case, the commutator does nothing : L R L' R' = L L' R R' = I

So in some way, the commutator allows you to measure to which extent the moves do not commute.

Effect

The sequence A changes a set of pieces located at some locations J on the cube. The sequence B affects some locations K. If J and K have no intersection, then A and B commute, and the commutator does nothing (for example [L, R]). Otherwise, J and K have an intersection N.

Pieces that are brought into the intersection by A are N A' (apply the inverse of A to the locations N). Pieces that are brought into the intersection by B are N B'.

The pieces that are affected by the commutator [A, B] = A B A' B' are located in the union of N, N A' and N B'. In other words, pieces that are affected by a commutator are those who are at the intersection of both moves, or are brought into the intersection by A or B. Other pieces, even if they are temporarily mixed up by A and B, will be put back at their original location with A' and B'.

Trivial case

When J and K have an empty intersection, A and B commute, so the commutator does nothing.

B inplace

If N B' = N, the sequence B only moves affected pieces that are inside the intersection. It may also move pieces that are outside the intersection, but those moves will be cancelled at the end. Affected pieces will only be N and N A'. So those pieces will be in J, i.e. among pieces that are directly affected by A. For example : [M', U2] = M' U2 M U2

In this example, N are the top-front and top-back edges. N B' is N U2, which is exactly N. N A' are front-bottom and front-top edges. So affected pieces are top-back, top-front, and front-bottom edges, which are directly affected by M and M'. In other words, affected pieces are in the middle slice.

In this case, it is relevant to consider that [A, B] = (A B A') B' = [A: B] B'. First part is the conjugate of B by A, and second part is the inverse of B.

In our example, the conjugated move is U2. Because only middle slice pieces are changed at the end, we can consider that it just swaps top-back and top-front edges. Thus, first part is M' U2 M swaps front-top and front-bottom edges.

So [M', U2] can be understood as : swap front-top and front-bottom edges, then swap top-back and top-front edges, which yields a 3-cycle of edges.

A inplace

This is a symmetrical case. Affected pieces will only be N and N B'. So those pieces will be in K, i.e. among pieces that are directly affected by B. For example, the inverse of [M', U2] which is (M' U2 M U2)' = U2 M' U2 M = [U2, M']. Again, affected pieces will be on the middle slice.

In this case, it is relevant to consider that [A, B] = A (B A' B') = A [B: A']. First part is A and second part is the conjugate of A' by B.

Other examples :

• A = "rotating a corner" like [R' D' R D R' D' R, U]
• A = "flipping an edge" like [R' E' R2 E2 R', U]
• A = "exchanging two corners" like [R' L' D2 R L, U]
• A = "exchanging two edges" like [M2 D2 M2, U]

Three different sets

When N A' and N B' are not included in N, there are three sets involved in the commutator :

• N A' : pieces that are brought into the intersection by A
• N B' : pieces that are brought into the intersection by B
• N : the intersection of J and K, i.e. where A and B interfere.

Note that N A' can have locations in common with N and N B' can have locations in common with N. In other words, it is possible that a sequence (A or B) does two different things in the context of a commutator :

1. moving pieces inside the intersection (from N to N)
2. bringing pieces into the intersection (from N A' \ N to N or from N B' \ N to N)

When there is only case 1, it is in fact the cases studied before. So we are left with those possible cases :

• Case 2 only : the three sets N A', N B' and N are completely separated (no overlap)
• Case 1 and 2 at the same time : the three sets overlap

No overlap

When sets do not overlap, the commutator can be summed up by :

• A stores the content of the intersection P_I in N A, and replace it with some pieces P_A from N A'
• B stores P_A in N B, and bring some pieces P_B from N B' into the intersection
• A' retrieves P_B from the intersection and places them in N A', and bring to the intersection previously stored pieces P_I in N A
• B' retrieves P_I from the intersection and places them in N B', and finally place P_A in the intersection.

Which can be described as :

step	intersection N	N A'	storage N A	N B'	storage N B
Initial	PI	PA	/	PB	/
After A	PA	/	PI	PB	/
After B	PB	/	PI	/	PA
After A'	PI	PB	/	/	PA
After B'	PA	PB	/	PI	/

So it is a 3-cycle of (P_I, P_A, P_B) into (P_A, P_B, P_I)

We notice that N A and N B are used as temporary storage :

• The content of the intersection, P_I, is first hidden by A in its storage N A, and comes back with A', so that the only transformation applied to these pieces is B'.
• The content of N A' is brought by A, and is then hidden by B in N B, so that the only transformation applied to these pieces is A.
• The content of N B' is brought by B, and then is placed immediately in N A' by A'. So the transformation applied to these pieces is B A'.

In other words, all the commutator do is :

• P_A goes A
• P_B goes B A'
• P_I goes B'

Examples

• [R' E2 R, U'] is a 3-cycle of edge, where P_A is the back-left edge, P_B is the top-front edge and P_I is the top-right edge.
• [R' D' R, U'] is a 3-cycle of corners

Overlap

This is the most complex case, which is a mix of the "no overlap" case and cases above (where N A' is included in N or N B' is included in N).

Example : [R, U]

See also

• Commutators and Conjugates
• Conjugate