From:

~~~ Subject:

I wrote in my e-mail message about the SKEWB of 1995/01/15

Here is the table for H. The first column contains the lenght. The

second column contains the number of positions of that length in H.

The third column contains the number of positions of that length that are

local maxima, i.e., the number of positions <pos> such that for no

generator <gen> the position <pos>*<gen> is longer. The fourth column

contains the number of positions such that for one generator <gen> the

position <pos>*<gen> is longer. And so on. So the eleventh column

contains the number of positions <pos> such that for all eight generators

<pos>*<gen> is longer (this happens of course only for the 2 solutions).length #pos #loc max 0 2 0 0 0 0 0 0 0 0 2 1 16 0 0 0 0 0 0 16 0 0 2 96 0 0 0 0 0 0 96 0 0 3 576 0 0 0 0 0 0 576 0 0 4 3456 0 0 0 0 0 240 3216 0 0 5 20496 0 0 0 48 729 2766 16953 0 0 6 118608 48 161 1231 4228 14779 32993 65168 0 0 7 630396 8266 33358 76349 121363 153892 137755 99413 0 0 8 2450966 1025322 621763 381098 234661 128570 47822 11730 0 0 9 2911712 2768641 126056 15344 1422 199 50 0 0 0 10 162056 161876 180 0 0 0 0 0 0 0 11 180 180 0 0 0 0 0 0 0 0

... note that this is the H = < RUF, RUB, LUF, LUB > ...

And in my e-mail message of 1995/01/31

I rerun the computation using the new subgroup H as a model for the

essential SKEWB. Here is the output.0 1 0 0 0 0 0 0 0 0 1 1 8 0 0 0 0 0 0 8 0 0 2 48 0 0 0 0 0 0 48 0 0 3 288 0 0 0 0 0 0 288 0 0 4 1728 0 0 0 0 0 120 1608 0 0 5 10248 0 0 0 36 377 1322 8513 0 0 6 59304 12 87 662 2217 7561 15698 33067 0 0 7 315198 4331 16897 37723 61161 76931 66997 51158 0 0 8 1225483 515249 311594 186221 115830 65096 25012 6481 0 0 9 1455856 1384909 61839 8280 708 95 25 0 0 0 10 81028 80938 90 0 0 0 0 0 0 0 11 90 90 0 0 0 0 0 0 0 0As you can see, the numbers in the first column are exactely half of the

corresponding numbers in my previous message (as was expected). The

numbers in the other columns are close to half of the corresponding

numbers in my previous message but not exactely identical. I have to

rethink what those numbers mean and how they relate to the corresponding

numbers for the basic SKEWB.

... note that this is now H = < RUF, LUB, RDB, LDF > ...

The reason that the numbers in the other columns of the second table are

not exactely half of the corresponding numbers in the first table is

rather simple. They are *both wrong*.

The correct numbers for H = < RUF, LUB, RDB, LDF > are as follows

0 1 0 0 0 0 0 0 0 0 1 1 8 0 0 0 0 0 0 8 0 0 2 48 0 0 0 0 0 0 48 0 0 3 288 0 0 0 0 0 0 288 0 0 4 1728 0 0 0 0 0 0 1728 0 0 5 10248 0 0 0 0 120 240 9888 0 0 6 59304 0 0 84 96 1740 6024 51360 0 0 7 315198 198 144 3600 9768 42900 94344 164244 0 0 8 1225483 15783 73016 199808 316776 343992 208584 67524 0 0 9 1455856 1001960 365792 74976 11760 1224 144 0 0 0 10 81028 80308 720 0 0 0 0 0 0 0 11 90 90 0 0 0 0 0 0 0 0

and the correct numbers for H = < RUF, LUB, RUB, LUF > are exactely twice

as large.

I figured out what those numbers mean. It is all rather simple.

Everybody who thought about them probably knows everything that follows.

I use the terms from my last few messages about models for the cube.

The basic states of cost 1 are exactely the elements in (F S F), where F

is the subgroup of essentially free elements, and S is the set of simple

elements (the set of generators) of G. Not all those elements need to be

different. Assume that there are <n_b> basic states of cost 1. Each

basic state <g> has <n_b> neighbors, namely the elements <g> (F S F).

The set of neighbors of each state is obviously a union of right cosets

of F. Furthermore if <g_1> and <g_2> are essentially equal, then there

sets of neighbors are equal. So we can map the whole concept to the

essential model G/F.

Recall that in the essential model G/F the set of elements of cost 1 was

exactely the set X = { (<x> F) | <x> in F S }. Assume that there are

<n_e> essential states of coset 1. Then each essential state (<g> F) has

<n_e> essential neighbors, namely the essential elements (<g> F) X.

We can now count how many of the basic neighbors of a basic state <g>

have smaller cost than <g>. If all <n_b> basic neighbors have smaller

cost than <g>, then we call <g> a basic local maximum.

Likewise we can count how many of the essential neighbors of the

essential state (<g> F) have smaller cost than (<g> F). If all <n_e>

essential neighbors have smaller cost than (<g> F), then we call (<g> F)

an essential local maximum.

It is easy to see that a basic element <g> is a basic local maximum if

and only if (<g> F) is an essential maximum.

In fact in most cases the number of basic states that have smaller cost

than <g> is simply (<n_b> / <n_e>) times the number of essential states

that have smaller cost than (<g> F). One sufficient condition for this

to happen is, that S is invariant under conjugation by S and that all

classes have the same length.

This condition is met for the SKEWB, so the numbers in the first table

*had* to be twice the numbers in the second table. Sorry about any

confusion I caused.

Have a nice day.

Martin.

-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany