From BRYAN@wvnvm.wvnet.edu Tue May 23 14:39:03 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA24587; Tue, 23 May 95 14:39:03 EDT Received: from wvnvm.wvnet.edu by MINTAKA.LCS.MIT.EDU id aa14844; 23 May 95 14:07 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3063; Tue, 23 May 95 14:04:04 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 6221; Tue, 23 May 1995 14:04:04 -0400 Message-Id: Date: Tue, 23 May 1995 14:04:03 -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