# Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial

I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative). If we wish to construct a 1-variable polynomial $A(t)$, we can project using $t_i \rightarrow t$.

On the other hand, the Alexander-Conway polynomial is defined by $$\nabla L_+ -\nabla L_- = (t^{1/2}-t^{-1/2}) \nabla L_0.$$

But these two polynomials differ by a factor of $(1-t)$ in case there are more than 1 component $$\nabla_L = \frac{A(t)}{1-t}=\frac{\Delta_L(t,\ldots,t)}{1-t} \text{ (up to multiplication by \pm t^k)}.$$

I figured this out experimentaly and am trying to find a reference to this fact if it is even true in general. In this post I asked about the existence of such a "projected" polynomial and now I am wondering about the difference with the Conway version of the polynomial.

• Possible duplicate of Multivariable vs single variable Alexander polynomial for links? – Marco Golla Jul 23 '18 at 0:26
• It's not the same. I am specifically searching for the difference between the two polynomials (the multivariable Alexander / Conway-Alexander) and am not just questioning the existence (well-definedness) of such a polynomial. – Jake B. Jul 23 '18 at 9:07
• The other question ended with "I guess there is a normalized version in one variable that respects the skein relation.", which to me sounds a lot like the question you're asking here. (And the two answers to your previous question address this.) – Marco Golla Jul 23 '18 at 9:15
• Oh, yes. My assumption back then was that they are equal, but now I am asking for a reference to the experimental fact that they are connected by division of $1-t$. I agree that they are similar questions, but I am really interested in the reference. Besides, the two answers (including the link) does not tell anything about the division, which should occur. – Jake B. Jul 23 '18 at 14:32

The relation is $$\Delta_L(t)\stackrel{.}{=}\Delta_L(t,...,t)(t-1)$$ (no need to take the normalised version). One reference is Proposition 7.3.10 of Kawauchi's "survey of knot theory". Arguably, a more conceptual understanding also follows from the use of Reidemeister torsion, see for instance Paragraphs 1.1 and 1.2 of Turaev's "Reidemeister torsion in knot theory".