I am studying Gathmann's algebraic geometry, and the very first example has left me absolutely baffled. In Example 0.1.1, he considers the equation $$C_n = \big\{(x,y) \in \mathbb{C}^2 : y^2 = (x-1)(x-2) \cdots(x - 2n)\big\} \subseteq \mathbb{C}^2$$ He explains that this is not given by taking two copies of $\mathbb{C}$ and pinching the points corresponding to the value $1,\ldots,2n$. He argues this by considering a path $x = re^{i \phi}$ and let $\phi$ run from $0$ to $2\pi$.
Fair enough, but then out of nowhere comes the conclusion: "The way to draw this topologically is to cut the two planes along the lines $[1,2],\ldots,[2n-1,2n]$, and to glue the two planes along these lines as in this picture.
I absolutely fail to see how this image connects up to the equation. If I were to take the loop with $r = 2.5$ on the upper plane of the left picture, I'd never switch between the two planes!
What am I misunderstanding?
N.B. I am aware of this question, which corrects some misleading points about Gathmann's example. But it is still not at all clear to me why the picture is the way it is.