I'm having trouble with the following past qual question: Let $\phi \colon S^1 \times \mathbb{D}^2 \hookrightarrow S^3$ be an embedding, where $\mathbb{D}^2$ is the open unit disk in $\mathbb{R}^2$. Compute the homology groups of $S^3-\phi(S^1 \times \mathbb{D}^2)$.
My thought was to consider $\mathbb{R}^3$ minus some solid torus, and realize the space in question as the 1-point compactification of this space. This, however, has not been terribly successful so far.
Thanks in advance for any help or hints!
As suggested by @Lee Mosher, using the Mayer-Vietoris sequence with $A = \phi(S^1 \times \mathbb{D}^2) \cup \{ small \textrm{ } nbhood \}$ and $B = S^3-A$, we get that $A \cup B = S^3$ and $A \cap B \simeq T^2$. If we want to use Mayer-Vietoris to solve for just the first homology group say, then the sequence $$ H_2(A \cup B) \to H_1(A \cap B) \to H_1(A) \oplus H_1(B) \to H_1(A \cup B) $$ becomes $$ 0 \to \mathbb{Z}^2 \to \mathbb{Z} \oplus H_1(S^3-\phi(S^1 \times \mathbb{D}^2)) \to 0, $$ which implies that $H_1(S^3-\phi(S^1 \times \mathbb{D}^2)) = \mathbb{Z}$.