[sent by Mike Reid (reid@math.arizona.edu), November 2002]
you might be interested in a "slick" proof for the skew
tetromino/aztec diamond:
let
/ 1 1 0 \ /-1 0 0 \
X = | 0 1 0 | Y = | 0 1 1 |
\ 0 0 -1 / , \ 0 0 1 /
then one can easily verify that
X^2 Y X Y X^-2 Y^-1 X^-1 Y^-1 = Y^-2 X Y^-1 X Y^2 X^-1 Y X^-1 =
X Y X Y^2 X^-1 Y^-1 X^-1 Y^-2 = Y^-1 X Y^-1 X^2 Y X^-1 Y X^-2 = identity,
but
/ 1 0 -4n \
(X Y)^2n (Y X^-1)^2n (X^-1 Y^-1)^2n (Y^-1 X)^2n = | 0 1 0 |
\ 0 0 1 /
which is non-trivial if n > 0 .
of course, this doesn't give much idea about *where* the proof came from!
but the tile homotopy group in this case can be completely analyzed.
mike
[see http://hedgehog.math.arizona.edu/~reid/Research/tilehomotopy.html
for more info]