The Hopf link is torus, but if I connect it with some arbitrary knot then what would the resulting link be? Is there some understanding of how it may relate to the class of the knot itself (torus/ hyperbolic/ satellite)? For example what if we take the connected sum of Hopf link with the hyperbolic figure-8 knot?
The literature on classification of Knots is plentiful, but for multi-component (or even 2) Links some references would be useful.