We can construct a covering space of a knot complement by cutting along a Seifert surface and glue several copies together. So I want to know how the complement of a Seifert surface looks like.
What I can only say is that the boundary of this space is a surface obtained by gluing two copies of the Seifert surface along their boundary.
For the unknot, I can easily imagine after cutting along a disk in $S^3$, we get a 3-ball because a 3-ball ($D^2\times I$) in $S^3$ is removed. How about more non-trivial examples? How about trifoil?