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. $B^4 \setminus \nu(D)$

where $\nu(*)$ denotes the tubular neighborhood. I understand that the answer is yes for slice knots.

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.