Is there a name for a 2-dimensional dumbbell like shape? Is there a mathematical name for a 2-dimensional shape in the general form of a dumbbell?  That is two circular nodes connected by a center beam such as shown in this image from this answer.  It could also be thought of as a circle that has been pinched together.  I believe topology does not help us here since there are no holes, and thus such as shape is the same as a circle, but perhaps geometry gives us a name for it?
The shape reminds of a Cassini oval (http://mathworld.wolfram.com/CassiniOvals.html) but without a more precise equation, one cannot be more specific.
You can try this

$$x^4-4(x^2-y^2+0.5)=0$$