Knot theory: Genus of a surface Use Euler characteristic to determine the genus of the surface in Figure 4.24 in picture below.
I am stuck with this question 4.10 from Colin Adams, the Knot Book.

I believe that you should first break up those faces a bit more, to make them polygonal. That mean, add an edge between any vertex on an inner square to the outer square in a given plane. There are 3 big faces, plus 12 big faces that have been cut in 4, plus 4 faces from the very small rod, 4 from the medium rod, 7 medium faces in the inner cube and 12 more trapezoidal faces after breaking up the remaining bad faces. 
Now count the edges. I believe there should be 84 of them, but it might be a good idea to recount them while drawing the diagram. 
Finally, no vertices were added in the polygonization, so there are still 40 of them. As the comments say, $\chi=V-E+F=40-84+42=-2=2-2g$, so the genus is 2. 
