Calculating the complement of a knot embedded in $S^3$ via Long Exact sequence of the pair $(S^3,K)$.
Let $K$ be a knot embedded in $S^3$. The following sequence of chain complexes is exact:
$0 \rightarrow C(K) \rightarrow C(S^3) \rightarrow C(S^3,K) \rightarrow 0$
Since this short sequence is exact, it induces the following L.E.S. In homology groups:
$0 \rightarrow H_3(S^3) \rightarrow^a H_3(S^3-K) \rightarrow^b H_2(K) \rightarrow^c H_2(S^3) \rightarrow^d H_2(S^3,K) \rightarrow^e H_1(K) \rightarrow^f H_1(S^3) \rightarrow^h H_1(S^3,K) \rightarrow^i H_0(K) \rightarrow^j H_0(S^3) \rightarrow^k H_0(S^3,K) \rightarrow 0$
I proved here: Use a Meyer-Vietoris sequence to calculate $S^3 - K$ where $K$ is an embedding of $S^1$. that the homology groups are trivial for $i>1$ and infinite cyclic for $i=0,1$.
I am now trying to reach the same result using a different long exact sequence. However, I ran into difficulties right away. The fact that $H_2(K) \cong H_2(S^1) \cong 0$ implies that $H_3(S^3,K) \cong H_3(S^3) \cong \mathbb{Z}$ which is not true. I feel like everything that I've done so far is pretty standard and was hoping that somebody could tell me what is going wrong here. Thanks!