# Exchanging the components of a two-component link

Given a 2-component link in $$S^3$$ whose components are trivial knots, is it always possible to find a homeomorphism of $$S^3$$ that exchanges the components?

I guess the answer is "no" (but I could not find a counterexample), so here is a second question:

Which link-invariant could prevent the existence of such an exchanging homeomorphism?

Indeed, as you suspect, the answer is no. For instance, take the link $$L$$ obtained from the Hopf link by doing a $$(2,1)$$-cable of one component and a Whitehead double of the other.
A reflection of this fact is that if you twist each of the components with respect to the other, you obtain different knots. (By "twist" here I essentially mean "do $$\pm1$$-surgery along one component.)