Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other defined using the hyperbolic geometry of the exterior. They are:
- The group of hyperbolic isometries of $S^3 \setminus K$ which admit continuous extensions, making them maps of pairs $(S^3,K) \to (S^3,K)$.
- The group of spherical isometries of the pair $(S^3,K)$.
I am curious if there are any natural ways to build on this, to make a thread of ideas. For example, is it possible to take a fibred hyperbolic knot, and put both the knot and all of the fibres (of the bundle) into a position that makes the fibres simultaneously minimal surfaces in both the spherical geometry and the hyperbolic geometry?
Any further analogies would be of interest to me but let's say the above question is the primary point of this thread.