From:

~~~ Subject:

Recall that Q* has been defined as the set of representatives

Q* = {F*, R*, L*, B*, U*, D*, F'*, R'*, L'*, B'*, U'*, D'*}

where * has been defined as a function selecting a representative

element of an M-conjugacy class.

I have done a little experimentation with cycles of the form X* ^ n.

As long as the X* are directly in Q*, the sequences are quite short,

and the final cycle is of length 2 in all cases. I found the latter

surprising initially, but with the wisdom of hindsight, it was

inevitable. Here is a table of lengths.

Operation Length of SequenceF* 11 U* 10 L* 7 R* 3 D* 9 B* 2 F'* 7 U'* 10 L'* 4 R'* 6 D'* 7 B'* 2

A couple of points of clarification:

1. As an example, for F*, we take i(F*)^n (that is, apply F*

to Start repeatedly). The sequence has 11 elements before it

repeats, then the 10-th and 11-th element repeat over and

over again. In all twelve cases, it is the last two elements

of the sequence which repeat.2. In order to replicate my results, you would have to define

a representative element function exactly like mine. Every

choice of representative element function can be expected to

yield different results.

To take a little more interesting case, I tried i(F*D'*) ^ n. In this

case, there were 63 unique elements in the sequence, and then the

8-th through the 63-rd elements repeated. Hence, the final cycle had

56 elements rather than the 2 elements of the simpler cases.

I suppose I could try quite a few other cases, but I have no idea how

to predict how long the sequences or the terminal cycles might be.

All I know to expect for sure is for things to be quite ill-behaved.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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