Let $E$ be a $n$-component string link in $D^2\times I$ with exterior $X$ and $X_0=X\cap D^2 \times 0$. Then the pair $(X,X_0)$ is $\mathbb Z$-acyclic and, given a representation $\rho :{\mathbb Z}[\pi ]\rightarrow \Lambda $, with $\pi =\pi _1(X)$, we use the property that this pair is $\Lambda $-acyclic for various representations $\rho $ and rings $\Lambda $ to construct torsion invariants for string links taking their values in the group $K_1(\Lambda )/ \rho (\pm \pi )$. A particular case is the Alexander polynomial in $n$ variables when $\Lambda $ is the ring of rational fractions $P/Q$ with $Q(1,1, \dots ,1)=1$ and $\rho $ is simply the abelianization map.