Boy's surface, visualization of the preimage of self-intersection locus as graph on projective plane

Question

For the immersion of the projective plane in $\mathbb{R}^3$ with one triple point, what does the preimage of the self-interaction locus as a graph on a projective plane look like?

Answer 1

So, I am not doing this very rigorously, but here it is.

This is Boy's surface, and from this angle, we can see the 3-fold symmetry that is present in it.

We see the triple point, right at the center. Since it is a single point in the immersion, there must be 3 points in the preimage, and we can this of this happening when we move the three sheets that meet each having a copy of the point. Now, that the triple point is kinda making sense, lets move to to the self-intersections of degree 2. The blue line connects the original red dot to itself. But there are 3 sheets and each sheet has a single degree 2 intersection with the other two, so we should have six edges in the preimage. (I am only drawing one edge here.)

If we do the same pull into different sheets trick we did for the points, we see that one edge connectes the red dot to itself. And pulling in the other direction, we have the blue lines connects the other two colors together.

And since there is 3-fold symmetry, this is what we should have for each of three dots.

So, we should get this graph, when we poke a hole in this surface and retract back to the identification "square." Each edge is colored to match which are identified in the imersion.

Hope this helps.