Date: Sun, 21 May 95 07:18:49 -0400
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ 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 <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
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