Question

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

Suppose that $K$ is an alternating knot.

CONJECTURE: The sequence of absolute values of the coefficients is unimodal. Specifically,

if , then

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

Answer 1

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.

Answer 2

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.

Answer 3

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.