So I am reading about essential surfaces, and I know that there is an essential torus in a punctured surface times $S^1$. I just don't see it? The only torus I can think of would be one $\partial$-parallel to the puncture times $S^1$. Where does it lie?
Secondly are there examples other than this and $F\times \{pt\}$ in the mapping torus of $F$, of closed essential surfaces in a $3$-manifold? I am thinking about broad categories like the examples I gave.
Thank you so much!
A punctured surface generally contains many different simple closed curved that are not "parallel to a puncture", i.e. not homotopic into an annulus of the surface surrounding the puncture; I'll refer to this as a "nonperipheral simple closed curve". There are a few exceptions to this, namely a sphere with 1, 2 or 3 punctures. But on surface $F$ which is a sphere with 4 or more punctures, or any other punctured surface whatsoever, there always exists a simple closed curve $c$ which does not bound a disc and is not peripheral, and so $c \times S^1 \subset F \times S^1$ is an essential torus.