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Use generators $a,b,c$ for the three one-handles in the generalized Heegaard diagram. Read a downward crossing of the cut from a left-hand hole to its right-hand partner as the corresponding generator. Traverse the attaching curve around the left-hand $A$ hole, starting at its outgoing downward strand, and then traverse the attaching curve around the left-hand $B$ hole in the same way. Their cut-crossing words areThese are the two relators imposed by the two handle attachments. The remaining boundary is a torus; no third relator should be added. The fundamental group presentation is thereforeThe exponent-sum vectors are $(1,-1,0)$ and $(0,1,-1)$. Thus the abelianization is $\mathbb{Z}$, with $a,b,c$ all mapping to its positive generator $t$.

Apply the Fox derivatives and then this abelianization. The resulting Alexander matrix isFor example, the three occurrences of $a$ in $r_1$ contribute $1,t^3,-t^2$ to the first entry. Every row sums to zero, as the Fox calculus identity predicts.

The cellular boundary in degree one for the infinite cyclic cover is $(t-1,t-1,t-1)$. Its kernel is freely generated by $e_a-e_c,e_b-e_c$. Expressing the two relation boundaries in this basis gives the two-by-two matrix obtained by deleting column $c$ of $F(t)$. Its determinant is the order of the Alexander module of a space. Equivalently, all three maximal matrix minors of $F(t)$ agree up to sign. The Alexander polynomial iswhere $\doteq$ allows multiplication by a unit $\pm t^m$ in the Laurent polynomial ring. Its value at $1$ is $1$.