Euler characteristic of Seifert surfaces

Question

For the above problem, I wanted to use the formula for surface-with-boundary with a single boundary component $\chi = 1-2g$, but I don't know how to find genus of those surfaces, can anyone help? This is in my knot theory course, sorry I don't know much about topology.

Answer 1

I would say it's much easier to do these computations by noting the graph to which each surface deformation retracts, and computing the Euler characteristic of that graph.

Just as an example, (b) clearly deformation retracts to a $\theta$ graph (although in this case you have to turn $\theta$ on its side). That graph has $2$ vertices and $3$ edges so $\chi = 2-3=-1$. And then, of course, using the equation $\chi = 1-2g$, you can compute $g = 1$.