I am teaching a course in knot theory, and I would like to describe the presentation of the Alexander module of a link obtained via Fox differential calculus. In doing this, I should prove the following fact.

Let $L$ be an $n$-component link in $S^3$ with complement $M$, and let $x_0\in M$ be a basepoint. Let $\widetilde M$ be the maximal abelian covering of $M$, and let $\widetilde{M}_0$ be the preimage of $x_0$ in $\widetilde{M}$. Fox differential calculus shows how to obtain a presentation matrix for $H_1(\widetilde{M},\widetilde{M}_0;\mathbb{Z})$ (with coefficients in $\mathbb{Z}[t_1^{\pm 1},\ldots, t_n^{\pm 1}]$) from a presentation of $\pi_1(M)$.

The usual proof of this fact uses cellular homology, and proceeds more or less as follows: 1. Once a finite presentation of the group $G$ is given, if $X$ is the $2$-dimensional cellular complex associated to the presentation (with only one $0$-cell, one $1$-cell for every generator and one $2$-cell for every relation), then the matrix associated to the presentation of $G$ provides a presentation of $H_1(\widetilde{X},\widetilde{X}_0;\mathbb{Z})$; 2. Using Tietze's Theorem, one shows that the module presented by the Fox matrix of a presentation of a group does not depend on the chosen presentation; 3. One shows that the link complement $M$ admits a realization as a cellular complex with only one $0$-cell.

In fact, point (3) allows to provide a presentation of $\pi_1 (M)$ whose Fox derivatives compute $H_1(\widetilde{M},\widetilde{M}_0)$, by point (1). Then (2) allows us to use any other presentation to get the same result.

My question is: is it possible to prove that $H_1(\widetilde{M},\widetilde{M}_0)$ is presented by the Fox matrix of any presentation of $\pi_1 (M)$ without relying on the fact that a link complement admits a cellular structure with only one $0$-cell (and maybe without using cellular homology too much)? I feel that there should be a proof that does not use cellular homology, and only relies on Hurewicz Theorem.