Is there a $4$-component link such that upon removing any one of them you get the Borromean link? Is there a $4$-component link such that upon removing any one of them you get the Borromean link?
I've managed to get close but not quite. What I have gets me something similar to the Borromean link but two of the components actually form what I think is the Whitehead link.
 A: You can sort of think of the Borromean rings as lying on three faces of a tetrahedron, with the center triangle of the usual presentation at a vertex.  By adding a component corresponding to the fourth face in a way so that the link has tetrahedral symmetry, you get the following:

While the outer component looks funny, it is the same as any of the other three, in the sense that there is an isotopy of the link sending this diagram to itself, but moving an inner component to the outer component.
A: This is essentially the same answer as that of Kyle, but with a more artistic aproach. If you vote this anwser, please vote Kyle's too since he's the one that inspired this design in the first place.

And here's the version with five components (two must be removed to obtain the Borromean Link).

A: Translating to Mathematica  the answer of Kyle, via the following diagram

we obtain the following picture

The corresponding Jones polynomial is

The corresponding Khovanov-Poincaré polynomial is

One application in quantum computing is as follows

