Remove one ring of Borromean rings in 3-sphere: linked or unlinked? We know Borromean rings in a 3-sphere $S^3$ can be unlinked if we remove one of the three rings.
Here let us consider a slight different procedure. If we remove the neighbored solid torus $B^2 \times S^1$ region of one of the three rings from $S^3$ , then the remained space after removing $B^2 \times S^1$ out of $S^3$ is still another $S^1 \times B^2$, since the gluing along gives $(B^2 \times S^1) \cup (S^1 \times B^2)=S^3$.
Question: Is the remained two rings out for the original Borromean rings are linked or unlinked in the remained complement space $S^1 \times B^2$ (which is the remained space from $S^3-(B^2 \times S^1)$)?
Note: $B^2$ is a 2-ball or equivalently a 2-disk. Say we remove the blue ring within $S^1 \times B^2$ and the remained red and green rings are still in the remained complement space $S^1 \times B^2=S^3-(B^2 \times S^1)$.

 A: As far as I understand the question, the answer is linked. You can't unlink the two remaining knots in the solid torus. An unlinking would give you an unlinking of the original problem, for which we know it does not exist.
A: By isotoping the link so that the red and green components are "obviously" in a solid torus defined by the blue component, we can get a better understanding of what the link is.

The red and green curves inside the solid torus look rather linked to me.
Rigorously, one may consider the case that there is a separating sphere within the solid torus, with one of the two components on each side. (Which, if there were one, would be a separating sphere in the original Borromean rings, which is probably what Daniel was referring to.)
Consider the infinite cover of the solid torus, and lift the link.

If there were a separating sphere in the solid torus, we could lift the sphere up to the infinite cover.  If the sphere contained, say, the red component (with the green component "outside"), then up in the cover it would still contain one of the lifts of the red component.  But, each green component links with the red component with linking number $1$, which would imply the green components must intersect this sphere, a contradiction.  Or in other words, there is a disk inside the lifted sphere whose boundary is the red arc, and the linking number implies a lifted green curve must intersect this disk -- and so the lifted green curve, and hence the green curve, is inside the sphere, too.
