Let $K \subset S^3$ be an arbitrary knot. Let $D$ denote the embedded disk in $B^4$ bounded by $K$.
Up to diffeomorphism, is it possible to describe the followings (at least for some trivial knots, such as trefoil, figure-eight knot,etc.):
1) $S^3 \setminus \nu(K)$
2) $B^4 \setminus \nu(D)$
3) $\partial (B^4 \setminus \nu(D))$
where $\nu(*)$ denotes the tubular neighborhood. I understand that the answer is yes for slice knots.