My question is this: Given an arbitrary (finite) presentation which maps onto $\mathbb{Z}$ by setting all the generators to be equal, do we get an Alexander Polynomial? Further, if there are more relators than generators, is the polynomail always zero? (I would guess Yes for the first, No for the second, but I think I read somewhere that you take it to be zero if there are more relators than regenerators, which contradicted something the author was saying elsewhere...thus my question!)
For example, the presentation $\langle a, b, c, x, y; ax=yb, ay=xb, bx=xa, by=yc, cx=xc, cy=ya\rangle$ has zero Alexander polynomial, while $\langle a, b, c; ab=ca, bc=ab, ca=bc \rangle$ has Alexander Polynomail $t^2-t+1$.