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Normalize the Alexander polynomial (in $t$) so that the positive and negative exponents are balanced. For example in the Conway normalization, make the substitution $z = t^{1/2} - t^{-1/2}$. The trefoil gives $t^{-1} - 1 + t$.

Conjecture: Suppose that $K$ is an alternating knot. Then the sequence of absolute values of the coefficients is unimodal. Specifically, if $\Delta_K(t) = \sum_i a_i t^i$, then $|a_0| \ge |a_1| \ge |a_2| \ge \cdots$.

This is a conjecture due to Murasugi, I believe. Where is it written? Has this been proved or disproved?

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    $\begingroup$ I think the knot has to be alternating. $\endgroup$
    – Autumn Kent
    Commented Nov 12, 2009 at 4:09
  • $\begingroup$ There's definately a lot of non-alternating counter-examples to this conjecture in the knot tables. $\endgroup$
    – Ryan Budney
    Commented Nov 12, 2009 at 4:12
  • $\begingroup$ Is it possible to rephrase this conjecture as about coefficients of Tutte polynomials of graphs? $\endgroup$
    – Sam Hopkins
    Commented Jul 9, 2022 at 20:49
  • $\begingroup$ There was recent progress on Fox's conjecture: arxiv.org/abs/2303.04733. (This paper is sort of along the lines of what I suggested in the previous comment - using a version of the Tutte polynomial for a certain kind of graphical representation of the knot, and then applying the powerful machinery of Lorentzian polynomials.) $\endgroup$
    – Sam Hopkins
    Commented Mar 9, 2023 at 1:53

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Didn't Hosokawa prove (1958, Osaka J. Math) that the Alexander polynomial of a knot can be any integral Laurent polynomial $p(t)$ such that $p(t^{-1}) = p(t)$ and $p(1) = \pm 1$?

If that's right, then according to Hosokawa, $2t^{-2}+t^{-1}-7+t+2t^2$ would be the Alexander polynomial of a knot, contradicting this conjecture.

It's been a while but I think you construct these knots very explicitly using ribbon diagrams -- Rolfsen's knots and links, also Kawauchi's big survey book should have the construction.

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  • $\begingroup$ Why is this an answer? The conjecture is about ALTERNATING knots... $\endgroup$
    – Igor Rivin
    Commented Jul 8, 2013 at 23:26
  • $\begingroup$ Alternating was edited-in after I posted this answer. As Sivek mentions, apparently the conjecture is still open. $\endgroup$
    – Ryan Budney
    Commented Jul 9, 2013 at 0:24
  • $\begingroup$ Ah, OK. It just looks a little strange, but maybe this exchange will deconfuse future generations. $\endgroup$
    – Igor Rivin
    Commented Jul 9, 2013 at 1:34
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It's actually a conjecture of Fox, sometimes known as the trapezoidal conjecture: the absolute values of the coefficients of $\Delta_K(t)$ are nonincreasing if K is an alternating knot. I think the original citation is Fox, "Some problems in knot theory."

Murasugi apparently proved the conjecture for alternating algebraic knots -- see "On the Alexander polynomial of alternating algebraic knots", MR0802722, which doesn't seem to be online -- and Ozsváth and Szabó proved it for genus 2 alternating knots in "Heegaard Floer homology and alternating knots," arXiv:0209149, but it's still open in general.

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  • $\begingroup$ Do you know why Fox made this conjecture? I wonder if there's a reason to suspect it, or if it's a correlation from loads of computations. $\endgroup$
    – Ryan Budney
    Commented Nov 12, 2009 at 4:57
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See also this paper by Jong which reproves the Ozsvath-Szabo result combinatorially, using Stoimenow's generators for knots of canonical genus 2.
The interesting question which is lurking in the background is the characterization of Alexander polynomials of alternating knots.

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