In "Topics in Differential Geometry" book of Peter W. Michor page 15, he gives an example of a compact surface of genus $g$ as follows:
Let $f(x) := x(x − 1)^2(x − 2)^2\cdots(x − (g − 1))^2(x − g)$. For small $r > 0$ the set $M:=\{(x, y, z) : (y^2 + f(x))^2 +z^2 = r^2\}$ describes a surface of genus $g$.
I tried to plot this example using MAPLE but I couldn't do it correctly (see wolfram for similar plot for $g=3$). My question is:
Q1: Is it possible to realize that this example is a surface of genus $g$ without plotting it?
Q2: Is there a nice pictured $g$-genus surface equation something like connected sum of $g$ torus?