 Solution

Write $X_{U,P}=S^3\setminus \mathrm{int}(\nu U\bigsqcup \nu K_P)$. Thenwhere the splice gluing exchanges meridian and longitude. Filling the remaining boundary along ${\mu }_P$ restores $\nu K_P$, so the pattern side becomes $X_U=S^3\setminus \mathrm{int}\nu U$, a solid torus. Under the splice gluing its meridian is attached to the meridian of $K_C$, because the meridian and longitude are exchanged twice in the two descriptions. This is the meridional filling of $X_C$, which restores the original three-sphere. Hence