Question: How is it possible to decrease the number of self-intersections in $\mathbb{R}^3$ while homeomorphically deforming the depiction of $N_2$ as the connected sum of two copies of $\mathbb{R}P^2$ into the depiction as the regular Klein bottle?
How does the connected sum of two copies of a surface which is difficult to immerse into $\mathbb{R}^3$ (i.e. $\mathbb{R}P^2$) become something which is easy to immerse into $\mathbb{R}^3$ (the Klein bottle)? Wouldn't we expect the connected sum to be "twice as difficult" to immerse, not half as difficult?
(Here the number of self-intersections necessary is our measure of "difficulty".)
Is the number of self-intersections perhaps a diffeomorphic invariant if not a homeorphic one?
Background: The typical depiction of the Klein bottle in $\mathbb{R}^3$ has only one self-intersection (or more precisely one curve of self-intersection, obviously not only one point).
Using the cross-cap depiction of the real projective plane (which is not an immersion), it has one self-intersection, so taking the connected sum $N_2$ of two such depictions, we should have two self-intersections.
(And if we took the connected sum of two copies of Boy's surface, then seemingly we would have even more than two self-intersections for the resulting surface homeomorphic to the Klein bottle, as this video shows.)
Obviously this implies that the number of self-intersections is to some extent not a topological invariant, since all of these surfaces are homeomorphic (the Klein bottle and $N_2$).
(Although this page says that these two different immersions of $\mathbb{R}P^2$ aren't homeomorphic? I definitely do not understand why that claim could be true.)
But at the same time, all of these surfaces clearly have a lower bound of $1$ for number of self-intersections, so seemingly the lower bound for number of self-intersections remains a candidate for a topological invariant.
Comment: This question of self-intersections may somehow be related to something called isotopy and higher-dimensional versions of knot theory. For instance, all knots which cannot be untied without self-intersections in $\mathbb{R}^3$ can be untied in $\mathbb{R}^4$, which resembles the properties of depictions of non-orientable surfaces in $\mathbb{R}^3$.
https://en.wikipedia.org/wiki/Homotopy#Isotopy
https://en.wikipedia.org/wiki/Regular_isotopy
https://en.wikipedia.org/wiki/Ambient_isotopy
https://en.wikipedia.org/wiki/Knot_theory