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 know it isn't using the Euler characteristic, but I couldn't help it. Consider the following 'proof by picture':
Start by rounding everything out, so that your original picture looks like a ball with holes drilled out of it. Then just follow the pictures.


This is an ambient isotopy of the manifold, so the genus is preserved, and it follows that the genus of your original surface is 2.
