From:

~~~ Subject:

On 20 May, I said the following:

I doubt that Mark's theory about GAP using C-conjugacy for slice

instead of M-conjugacy is correct. I already have 50 positions

to 23 for GAP, and using C-conjugacy would just make my results

larger. For example, RL' and R'L are M-conjugate positions,

but not C-conjugate positions.

We already know from Martin Schoenert that GAP is using neither

M-conjugacy nor C-conjugacy, but <slice>-conjugacy. But my

statement about C-conjugacy vs. M-conjugacy was completely incorrect

in any case.

Dan Hoey pointed out to me that RL' and R'L in fact *are* C-conjugates

under 180 degree rotation around the U-D axis.

We can observe that R and R' are not C-conjugates, nor are L' and L,

which suckered me into stating that RL' and R'L are not. But

rewrite R'L as LR' since opposite face moves commute. Now, RL'

and LR' are clearly C-conjugate.

In fact, I have now verified with a quick search program that

all M-conjugates in the slice group are also C-conjugates. Hence,

there are 50 C-conjugate classes in slice, just as there are

50 M-conjugate classes.

In retrospect, I don't think the search program was necessary.

Suppose X and Y are M-conjugates in the slice group. Then,

they can be written as M-conjugate sequences. That is, they

can be written so that the individual slice moves are respective

M-conjugates for some fixed m in M. (The fact that it might be

possible also to write them so that the individual slice moves

are not respective M-conjugates for some fixed m in M is

irrelevant.) Furthermore, write the sequence for X so that the

clockwise half of each slice is written prior to the

counter-clockwise half of the slice. The sequence for

Y with individual slice moves being respective M conjugates of

X may or not have this property. But if not, then simply reorder

the halves of the slices of Y to put the clockwise half first, and Y

will still be piecewise M-conjugate with X. Then, the piecewise

M-conjugate slices are also C-conjugate, and therefore the X and

Y positions are C-conjugate.

Antislice is totally different. For example, RL is M-conjugate to

R'L', but it not C-conjugate.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU