4
$\begingroup$

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.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Possible duplicate of Multivariable vs single variable Alexander polynomial for links? $\endgroup$
    – Marco Golla
    Jul 23, 2018 at 0:26
  • $\begingroup$ 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. $\endgroup$
    – Jake B.
    Jul 23, 2018 at 9:07
  • $\begingroup$ 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.) $\endgroup$
    – Marco Golla
    Jul 23, 2018 at 9:15
  • $\begingroup$ 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. $\endgroup$
    – Jake B.
    Jul 23, 2018 at 14:32

1 Answer 1

Reset to default
4
$\begingroup$

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".

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.