Hatcher's exercise 1.2.12 consists of a few parts. I was able to solve all of them except the last part, which asks the reader to show that the immersed Klein bottle in $\mathbb{R}^3$, shown in the image on the left, has the fundamental group isomorphic to that of the complement of a certain graph in $\mathbb{R}^3$, shown in the middle.
This question was already asked here here but it wasn't given a full answer and the question was marked as solved without the part that's relevant to me being answered. One person indicated that these spaces have the same homotopy type but did not explain why.
My idea was to try to show they're homotopy equivalent spaces by showing that the complement of the graph deformation retracts onto the immersed Klein bottle. However, this runs into a difficulty as follows.
I try to imagine small loops around the graph and think where they would reasonably end up. This is illustrated in the images below. As we go up the graph to the outer loop, we enter a bifurcation point, and the loops past the bifurcation point seem quite inconvenient. It seems like there's no good place to map them without the ring intersecting the graph as it deformation retracts. So intuitively it seems to me like the space does not deformation retract onto the Klein bottle.
Is this correct? Are you supposed to demonstrate homotopy equivalence in some other way? Or are you supposed to try to compute a presentation for the fundamental group of the complement of the graph directly and see that the presentations give isomorphic groups? I tried that and couldn't get very far as well. However, I'm primarily interested in seeing whether there is a nice geometric argument for why the fundamental group should be isomorphic, since intuitively the graph seems closely related to the Klein bottle, I just don't understand how exactly.
