EDIT - I've rewritten my previous answer in an attempt to remove everything except the answer to your question.  All submanifolds are assumed to be smooth. 

> Suppose that $M$ is a closed, connected, oriented, irreducible three-manifold (and $M$ is not the three-sphere).  Suppose that $K$ is a knot in $M$.  Then $\pi_2(M - K)$ is non-trivial if and only if $K$ is contained in an embedded three-ball $B^3 \subset M$. 

The proof is an exercise using Alexander's theorem (every embedded two-sphere in $S^3$ bounds balls on both sides), the [sphere theorem][1], and the [Poincaré conjecture][2].  Hempel's book and Hatcher's notes on three-manifolds are standard references for the necessary background material. 


  [1]: http://en.wikipedia.org/wiki/Sphere_theorem_%283-manifolds%29
  [2]: http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture