= Solution
Let $X_\# $ be the connected-sum exterior and fill it along its <Seifert longitude>
$$
\lambda_\#=\lambda_1\mathbin\#\lambda_2.
$$
The decomposing annulus in $X_\#$ has two parallel boundary circles, each meeting $\lambda_\#$ once. Applying part (b) cuts the filling back into $X_1$ and $X_2$ and identifies the peripheral curves by
$$
\lambda_1\longleftrightarrow\lambda_2,
\qquad
\mu_1\longleftrightarrow\mu_2,
$$
up to the signs required by the orientation-reversing torus gluing.
In $Y_1'=X_1(\lambda_1)$, the core $K_1'$ has meridian $\mu_1'=\lambda_1$. By the orientation stipulated in the question its longitude is $\lambda_1'=\mu_1$. Thus the displayed gluing sends the meridian and longitude of $K_1'$ to the longitude and meridian of $K_2$, respectively, which is exactly the <splice of knots>. Therefore the required integer surgery is zero surgery on the connected sum:
$$
\boxed{Y_\#'=X_\#(\lambda_\#)\cong
\operatorname{splice}((Y_1',K_1'),(Y_2,K_2)).}
$$
Back to article page
