Consider a connected 3D-printable shape such as the below. It appears that any plane passing through the centroid will divide the shape into more than two pieces. Define a shape with this property unhalvable.
Would the simplest unhalvable shape be topologically equivalent to a sphere, a torus, or something else?
What are some simple unhalvable shapes?
Is the below embedding of the Trefoil knot unhalvable?
I'd venture that it would be a sphere. Take a shape like the one you have above in $\mathbb{R}^3$ (which is topologically equivalent to a torus) and cut it somewhere. I guess that it's still "unhalvable", but is now topologically equivalent to a sphere.
One example of an unhalvable shape which I can think of is the co-ordinate axis in $\mathbb{R}^3$.
Whether the trefoil knot is unhalvable would seem to depend on how you embed it in $\mathbb{R}^3$ - because that would change where your plane intersects it. You probably would need to define "unhalvable" a little more rigorously to ask questions like that.