The genus of a Seifert surface is (1 - s + c)/2

We know that X = s - c from the previous excercise. Also, we know that a Seifert surface has exactly one boundary component, so if we cap it off, we will have a surface without boundary whose Euler characteristic is s - c + 1. The genus of the Seifert surface will be the same as the genus of this new surface without boundary. So we need to know the relationship between Euler characteristic and genus for surfaces without boundary. Since Seifert surfaces are orientable, we are interested only in the surfaces with even Euler characteristic (sphere, torus, double torus, etc.). All the surfaces with odd characteristic are homeomorphic to the real projective plane with zero or more handles, and therefore are not orientable. Here is a chart of Euler characteristic and genus of surfaces without boundary:

X	2	0	-2	-4	...
g	0	1	2	3	...

Note that as the Euler characteristic decreases by two, the genus increases by one. This is because whenever we add a handle to a surface, we remove two disks (X goes down by two) and add a hole (genus increases by one).

We see from the chart that for surfaces without boundary, g = (2 - X)/2. So, the genus of a Seifert surface is 2 - (s - c + 1)/2, or (1 - s + c)/2.