From BRYAN@wvnvm.wvnet.edu Sun May 21 15:24:47 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26938; Sun, 21 May 95 15:24:47 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0109; Sun, 21 May 95 07:18:50 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 3804; Sun, 21 May 1995 07:18:50 -0400 Message-Id: Date: Sun, 21 May 1995 07:18:49 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: AntiSlice Under M-conjugacy (and a problem with slice) I have had some posts not get through. The following will serve to consolidate several of them. Some of this may be a repeat, but not all, I think. Mark Longridge's antislice results are as follows: > arrangements arrangements >Moves Deep (2q or anti-slice moves) (4q or double anti-slice moves) > > 0 1 1 > 1 6 9 > 2 27 51 > 3 120 265 > 4 423 864 > 5 1,098 1,785 > 6 1,770 2,017 > 7 1,650 1,008 > 8 851 144 > 9 198 > ----- ----- > 6,144 6,144 We have the following M-conjugacy results for 2q moves. Level Positions Local Maxima 0 1 0 1 1 0 2 3 0 3 10 0 4 37 0 5 93 1 6 166 2 7 147 7 8 89 12 9 21 21 ---- 568 The level 5 local maximum is (U'D')(FB)(FB)(UD)(L'R'). The position is not its own inverse, but we can use as an inverse (U'D')(FB)(FB)(UD)(LR). Hence, (U'D')(FB)(FB)(UD) forms a nice "middle" of the sequence. In fact, the (U'D')(FB)(FB)(UD) position in some ways seems more interesting than the local maximum itself. Does it already have a name? I have not verified if the length of the local maximum is 10q in G, nor if it is a local maximum in G. We have the following M-conjugacy results for 4q moves. Strong and weak local maxima are defined according to my preference. If you prefer Mike Reid's definition, ignore the "weak" column and read the "total" column as "weak". Level Positions Strong Weak Total Local Max Local Max Local Max 0 1 0 0 0 1 2 0 0 0 2 5 0 0 0 3 25 0 1 1 4 75 0 2 2 5 152 0 19 19 6 184 1 35 36 7 108 0 46 46 8 16 0 16 16 ---- 568 Back on the subject of the slice group, we have the following. Mark Longridge said: >By the way GAP gives NumberConjugacyClasses (slice) = 23 >In your calculations of M-conjugacy classes for the slice group the >total number of classes is 50, but I think GAP does not use >M-conjugates but C-conjugates instead. The NumberConjugacyClasses >function always thrashes with any larger groups unfortunately. >If you could easily tweak your program perhaps you could >verify my theory. Recall that in my work with , I had to use W3-conjugacy rather than M-conjugacy. The simplest explanation is that all M-conjugates of a position in are not in , and in particular the representative element might not be in . W3 is the largest subgroup of M such that all conjugates of are in . I flirted briefly with the notion that I might have the same problem with the slice group and the antislice group. But it seems immediate that M-conjugacy is appropriate for both slice and antislice. For example, think of applying M-conjugacy to all the individual 2q or 4q moves in a slice or antislice process. Clearly, the result is still in slice and antislice, respectively. 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. I don't have a clue why my results do not match GAP. I have double and triple checked my results, and they seem correct. For example, I can "expand" my conjugacy classes, and the results then match Mark's exactly. How does GAP's NumberConjugacyClasses function work? By that, I mean how does it know the subgroup with respect to which you are taking conjugacy classes (if my terminology is correct)? For example, how does it know to take C or M or whatever conjugates? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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