From:

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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 Maxima0 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 Max0 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 <U,R>, I had to use W3-conjugacy rather

than M-conjugacy. The simplest explanation is that all M-conjugates

of a position in <U,R> are not in <U,R>, and in particular the

representative element might not be in <U,R>. W3 is the largest

subgroup of M such that all conjugates of <U,R> are in <U,R>.

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