Let $K$ be a knotted solid torus in $S^3$, $T$ be its boundary torus, and $X$ be its exterior, i.e. the closure of $S^3\setminus N$. The goal is to compute $H_n(X)$, the integral homology of $X$.
Using Mayer-Vietoris, it follows immediately that $H_n(X) = 0$ for $n \geq 4$ and $H_1(X) = \mathbb Z$. It remains to compute $H_2(X)$ and $H_3(X)$.
From Mayer-Vietoris, we have
$$0 \rightarrow H_3(X) \rightarrow H_3(S^3) \xrightarrow{\partial} H_2(T) \rightarrow H_2(X) \rightarrow 0$$
I was told that $\partial : H_3(S^3) \rightarrow H_2(T)$ is an isomorphism. With this fact, it is easy to show that $H_2(X) = H_3(X) = 0$ using exactness and the splitting lemma.
I would appreciate any help/hint with proving that $\partial$ is an isomorphism.