Let $K$ be an oriented knot in an oriented $3$-manifold $M$. Let $\nu(K) \approx S^1 \times D^2$ denote the tubular neighborhood of $K$ in $M$.
If $K$ is null-homologous, i.e., $[K]=0$ in $H_1(M)$, then why this implies that $$H_1(M \setminus int(\nu(K))) \cong \mathbb Z \oplus H_1(M).$$
P.S. I applied the Mayer-Vietoris argument to the pair $(\nu(K), M \setminus int(\nu(K)))$, so we have the following sequence: $$\ldots \to \underbrace{H_1(S^1 \times S^1; \mathbb Z)}_{\cong \mathbb Z ^2} \to \underbrace{H_1(\nu(K); \mathbb Z)}_{\cong \mathbb Z} \oplus H_1(M \setminus int(\nu(K); \mathbb Z) \to H_1(M; \mathbb Z) \to \ldots$$
I cannot effectively use the assumption that we have a null-homologous knot $K$ in $M$.