Conjugates

So, to recapitulate, we have a monoflip x, and we've constructed the commutator [x,U] that flips the upper front and right edges. In just the same way, we can construct the commutators [x,U^k] (for k = 2 and −1) that flip other edges on the upper face. But, what if we want to find a sequence that flips, say, the upper front and lower right edges? The monoflip x ties up the entire bottom layer, so we can't just pick some sequence y that puts the lower right edge in the hot seat and then construct the commutator [x,y]. What we can do, though, is take one of the commutators we already have, say [x,U], and think of it as a single indivisible sequence that has two hot seats and doesn't tie up anything! In the present case we can just add a g = R² at the start to put the lower right edge in the upper right hot seat and a g⁻¹ = (R²)⁻¹ = R² at the end to put it back, and voilà! In fact, with an appropriate value of g we can use a sequence of the form g[x,U]g⁻¹ to flip any two edges anywhere on the cube!

A sequence of the form ghg⁻¹ is called a conjugate of h. I'm going to go further and call it the conjugate of h by g, but that's not standard; many mathematicians would guess I meant g⁻¹hg. Like commutators, conjugates come from group theory, but unlike commutators, they're tremendously useful there. All the books except [WW] discuss conjugates.

Although it's true that we can flip any two edges by conjugating [x,U] by an appropriate value of g, in practice that's not the best approach. We have one degree of freedom in the value g, but there are two other degrees of freedom that we should also make use of. First, we get to choose the commutator. In other words, the degree of freedom is the value y in the commutator [x,y]. Here, in addition to the sequence [x,U] that flips adjacent edges on a face, we also have the sequence [x,U²] that flips opposite edges. ([x,U⁻¹] is redundant.) Second, we get to choose the orientation of the cube. This degree of freedom is very important! If we want to flip two randomly chosen edges, with probability 6/11 we can reorient the cube so that the edges are in place for one of the unconjugated commutators, and the rest of the time we can arrange that the value g need only contain a single turn.