Suppose we are flatlanders who want to test whether we live in $S^2$ or $T^2$. To do this, we set out in a certain direction, unraveling a rope as we walk. When we return to our starting point, we try to reel the rope back in. If the rope fails to reel in, i.e. it gets "caught on the space," then we know we live in $T^2$. If we can reel it in, we know we are in $S^2$. If we repeat the experiment, starting in a direction $90^{\circ}$ to the original direction, we should get the same result for the same space. ${\bf Question \ 1}:$ Is there a 2D space that is "in between" these two? A space in which one direction the rope can be reeled back in and another direction where it can't? Obviously the experiment wouldn't work if we lived in $S^1 \times I$, because we would hit a boundary.
Now let's extend this experiment to 3-manifolds. We can assume we are speculating about the shape of our own universe. In particular, I am interested in manifolds associated with physical rotations and their parametrizations: $T^3$, $S^3$, $\mathbb{R}P^3$, $S^2 \times S^1$, and so on. If you haven't heard of ${\bf the \ Balinese \ plate \ trick}$ or ${\bf the \ belt \ trick}$, I would recommend taking a look for the sake of this conversation. Configurations of belt twists are in a 1-1 correspondence with points of $S^3$, i.e. they are uniquely parametrized by unit quaternions. $S^3$ is also a manifold which allows us to reel in our rope, i.e. it is genus 0. Physically, this means that when the belt has two twists, it may be smoothly deformed back to its original configuration. This is the process of "reeling back in the rope." ${\bf Question \ 2}:$ Can anyone think of a physical example that demonstrates a rope getting caught on the space? To me this would seem to be some sort of physical system which returns to its original configuration but in an "irreversibly damaged" state.
Thank you.