I am trying to show that if $K\subset S^3$ is a $(p,q)$ torus knot, then the knot exterior $X_K=S^3\setminus N(K)$ is Seifert-fibered space, where $N(K)$ is a tubular neighborhood of $K$ in $S^3$.
It's obvious to me that the base space of such a fibration should have connected boundary, since $X_K$ has torus boundary. So, I've tried to define such a fibration on a disk (maybe by taking the standard fibered torus $D^2\times S^1$, and replacing a regular fiber from the interior with a singular fiber of some order depending on $(p,q)$). However, I don't know how to argue such a construction. Even if I can manage to make the right fiber replacement, I don't know how to show that the resulting space would be homeomorphic to $X_K$.
Can someone give me a hint about this?