Consider a knot K in the 3-sphere, its knot complement C(K) is obtained by removing a tubular neighborhood of K from the 3-sphere. What can I say about the complement of C(K) in the 3-sphere? Is it always diffeomorphic to some knot complement C(K')? And if so, is K' unique? I don't expect so, but maybe there's a canonical way to find one representative in the class of knots that can work as K'...
As a trivial example, the Hegaard splitting divides the 3-sphere into two solid tori, glued along their $T^2$ boundary, so K in this case is the unknot and K' too.
Addendum: I think you can prove that K' is the knot obtained from K via a 1-point inversion of the 3-sphere around a point that is not on the knot. Is it known what such an operation does to a knot?