Let $K$ be a knot in $S^3$, and N(K) be its tubular neighborhood. I want to compute the homology of $S^3-N(K)$ using Mayer-Vietoris. Let $A$ be $N(K)$ minus a small neighborhood, and $B$ be the complement of N(K) with a small neighborhood added. Note that $A\cup B$ is $S^3$, and $A\cap B$ deformations retracts onto a torus. $H_1(S^3-N(K))$ and $H_n(S^3-N(K))$, $n>2$, pop out of Mayer Vietoris applied to A and B. However I am struggling with $H_2$. According to Mayer-Vietoris we have the following exact sequence $$H_3(S^3) \to H_2(T) \to H_2(A)\bigoplus H_2(B) \to 0$$ The part I don't understand is the map $H_3(S^3) \to H_2(T)$. Let $\alpha$ be the generator of $H_3(S^3)$. Under the boundary map of Mayer-Vietoris, $\alpha$ is subdivided into a part residing in $A$ and a part residing in $B$. Then we take the boundary of the part residing in $A$. Thinking of the boundary map like this makes me think that it is an isomorphism. However, I am not sure of this.
Are there other ways to see what this map does to the generator of $H_3(S^3)$? More generally, how does one work with the boundary map in the MV sequence in higher dimensions, where things cannot be simply visualized?