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Date: Tue, 23 May 1995 08:50:40 -0400 (EDT)
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: M-conjugacy vs. C-Conjugacy in Slice and Antislice
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 -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