Is Murasugi's conjecture still open? 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?
 A: 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. 
A: 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.
A: 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.
