I would like to find a completely algebraic argument to the fact that the second homology group of a knot complement is trivial.
Concretely, let $K$ be a knot and let $V$ be a tubular neighbourhood of $K$ homeomorphic to the solid torus, and let $X:= \overline{S^3 - V}$ the knot complement. Using the Mayer-Vietoris sequence with the cover $U_1 := V$, $U_2:=X$ of $S^3$, we obtain the exact sequence
$$0 \to H_3 (S^3) = \mathbb{Z} \overset{\delta}{\to} \mathbb{Z}=H_2 (S^1 \times S^1) \to H_2 (X) \to 0 .$$
I want to find an algebraic argument (that is, avoiding sketchy arguments involving simplices like this) of the fact that the connecting homomorphism is an isomorphism. I tried to argue with the naturality of the long exact sequence, but with no success so far.