I was recently thinking about links where each component plays the same role: for every permutation of components, there is an isotopy permuting these components in the prescribed way. In the vein of knot/link invariants, we might ask how to tell when this is not the case. The obvious way to do this is by removing components and comparing the isotopy classes of the resulting links. However, this may be insufficient, particularly for Brunnian links. Are there any link invariants that treat components differently, so that they could detect if a Brunnian link does not have this permutability property?
Disclaimer: while I am very interested in topology, my knowledge of knots and links is very limited. This is also my first post to MO, so I’m happy to hear feedback like “you should’ve phrased your question this way” or “this is better suited to math SE.”