# Multivariable vs single variable Alexander polynomial for links?

If we take a $n$-component link $L$, we have the multivariable Alexander polynomial $\Delta(L)(t_1,\ldots,t_n)$. Is there a standard single-variable Alexander polynomial? If yes, is it just euqal to $\Delta(L)(t,\ldots,t)$? I guess there is a normalized version in one variable that respects the skein relation.

Both polynomials arise by considering the first homology of covering spaces of the link exterior $$X_L$$. If the link $$L$$ has $$n$$ components, then $$\Delta_L(t_1,...,t_n)$$ is extracted from the free abelian cover of $$X_L$$. The one variable polynomial comes from the infinite cyclic cover corresponding to the kernel of the map $$\pi_1(X_L) \stackrel{ab}{\to} H_1(X_L)\to \mathbb{Z}$$ that sends each meridian to $$1$$. The one variable version is for instance thoroughly discussed in Chapter 6 of Lickorish's "Introduction to knot theory". The relation $$\Delta_L(t)\stackrel{.}{=}\Delta(t,...,t)(t-1)$$ holds.

I think this is the one you get from HOMFLY polynomial. Recall that for the usual Conway triple $(L_+,L_-,L_0)$ HOMFLY polynomial $P$ has this relation:

$lP(L+)+l^{-1}P(L_-)+mP(L_0)=0$

Setting $l=i$ and $m=-i(t-t^{-1})$ would produce Alexander polynomial.

Also recall that Reshetikhin-Turaev construction of HOMFLY-equivalent polynomial family $\text{v}_n, n\ge2$ from $U_q(\mathfrak{su}(n))$ has this relation:

$q^n \text{v}_n(L_+)-q^{-n} \text{v}_n(L_-)=(q-q^{-1})\text{v}_n(L_0)$

$\text{v}_2$ happens to coincide with Jones polynomial, and $\text{v}_0$ is equivalent to the single-variable Alexander polynomial (after variable changes). Well, strictly speaking $v_0$ is not is not derived directly from Reshetikhin-Turaev construction because of the lack of $\mathfrak{su}(0)$, but one can show that the case of $n=0$ follows from the validity of $v_n,n\ge2$

Yes, there is such a one-variable polynomial. In fact, in my understanding, this came earlier than the multi-variable version.

Gwénaël Masseyeau has some notes on the subject, with references to work of Conway (for the Conway polynomial and skein relations), Kauffman (for fixing Conway's approach in the single-variable case), and Hartley (for the multi-variable case). Right before Example 2.10 he has the identity you're looking for.