 Solution

The component knot types can change under this Conway mutation. To construct an example, place a knotted arc carrying a trefoil knot summand in one strand of $D_1$ and a knotted arc carrying a figure-eight knot summand in one strand of $D_2$, leaving the other two strand portions trivial. Choose the outside pairing so that before rotation the two knotted portions lie on different components, while after rotation they lie on the same component. Then the component multisets arerespectively. A link isotopy preserves the unordered multiset of component knot types, so these links are not isotopic, although part (a) shows that suitable orientations give them the same Jones polynomial.