Suppose $M$ is a connected, closed $2$-dimensional manifold embedded in $\mathbb{R}^3$, and consider the set $I$, which is the intersection of $M$ with some plane $P$.
Can the $I$ always be described in terms of Jordan curves in any neat way? While generally the intersection will be describable as a disjoin union of Jordan curves, there are some degenerate cases where the intersection is composed of multiple Jordan curves that share an edge, so it might instead make sense to consider the set of minimal Jordan curves that lie in the intersection.
Is there any way to describe the number of Jordan curves and their containment relation (the partial ordering of one curve being contained in the interior of the other curve) that lie in the intersection of $I$?
Some examples:
If $M$ is convex, it's quite easy to see that the intersection with a plane is either empty, or a single loop, and similarly, I'd think (though I'm not sure how to show this) this is also true if $M$ has positive Gaussian curvature everywhere.
If $M$ is the torus, we can slice the torus "horizontally" so as to get two loops, one contained in the other. We can also slice "vertically" and get two loops that are non-intersecting. In addition to that, we can slice it so as to create a single loop, like the convex case above.
The genus of the torus doesn't seem relevant - the case of creating two loops with one in the other can be achieved if we slice a handless cup, and the case of creating two loops that do not intersect can be created if we slice a bean-like shape.
What about the general case? What are some tools or approaches that could be used to describe this problem? E.g. something that would at least formalize the examples I've given above. I'd appreciate reference that would cover this topic.