Arrgh. Corrections:
On Apr 30, 2015, at 12:03 PM, Dan Asimov <asimov@msri.org> wrote:
Consider the following graph in 3-space:
Start with a white rectangle, say [0,3] x [0,1].
Blacken the intervals {n} x [0,1] for n = 0, 1, 2, 3.
Also blacken the edges {0} x [0,1} and {1} x [0,1].
Also blacken the edges [0,3] x {0} and [0,3] x {1}.
Now identify (0,t) with (1,1-t) for all t in [0,1], et voilà: a Möbius band.
Now identify (0,t) with (3,1-t) for all t in [0,1], et voilà: a Möbius band.
Arrange this surface in space to be the usual picture of a Möbius band with one half-twist.
Finally, delete everything that is not blackened, leaving a graph of 6 edges and 6 vertices with a particular embedding in 3-space.
Call the abstract graph K, and its embedding in space the function
h: K -> R^3.
PUZZLE: -------
Can K be continuously manipulated through homeomorphic graphs in 3-space so that it coincides with its mirror image?
(Or more formally: Is there a continuous mapping
H: K x [0,1] -> R^3
such that
a) H(x,0) = h(x), for all x in K;
b) H restricted to K x {t} is an embedding into R^3, for all t in [0,1];
c) H restricted to K x {1} is an embedding into R^3 whose image is a mirror image K' of K
?)