It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$.

Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial M$ and $\partial N$ are homeomorphic. Suppose those boundary are a surface with genus $g$.

We fix a genus $g$ handlebody $H$ in $\mathbb{R}^3$. Let $f:\partial H \to \partial M$ and $g:\partial H \to \partial N$ be homeomorphisms (with a good choice of orientation preserving or reversing.)

Now we make closed oriented 3-manifolds $M\cup_f H$ and $N\cup_g H$. By the Theorem I stated at the beginning, there are framed links $L_1$ and $L_2$ that represent $M\cup_f H$ and $N\cup_g H$ respectively.

Next, we can construct another closed oriented 3-manifold $M\cup_{g^{-1}\circ f} N$.

My question is; Is there any way to get a surgery link for $M\cup_{g^{-1}\circ f} N$ from $L_1$ and $L_2$?


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