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 $t^{-1} - 1 + t$.



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> **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 $|a\_0| \ge |a\_1| \ge |a\_2| \ge \cdots$.
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This is a conjecture due to Murasugi, I believe. Where is it written? Has this been proved or disproved?



Is Murasugi's conjecture still open?
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. 


