Unknot is fibred

Question

I think I have understood the basic defintion of a fibred knot, but I am still unable to apply it to trivial examples, for example the unknot. Of course I could apply Milnor's theorem and (probably after a coordinate change) note, that for $f=x^2+y^2-1$ the map $\frac{f}{|f|}$ is a possible fibration, but I am more interested in getting an intuition , i.e. to geometrically see how and which Seifert surfaces of $S^1$ are parametrized by $S^1$. Thank you for your help.

Answer 1

Think of the unknot in $S^3$ as the $z$-axis in $\Bbb R^3$. Then the surfaces are the linear half-planes bounded by z. The picture looks like what would happen if you took a book, stood it up, and completely opened it up so that the pages pointed out at different angles.

If you really want to think about this in the standard unknot in $\Bbb R^3$, it starts with the obvious disc, then the disc moves upward to look like a spherical cap; eventually it looks like most of a very large sphere with a small cap below the unknot removed; eventually it becomes the complement of the disc in the xy-plane (still a disc once you include the point at infinity). Then you reverse this process in the lower half-space.