Didn't Hosokawa prove (1958, Osaka J. Math) that the Alexander polynomial of a knot can be any integral Laurent polynomial p(t)$p(t)$ such that p(t^-1) = p(t)$p(t^{-1}) = p(t)$ and p(1) = \pm 1$p(1) = \pm 1$?
If that's right, then according to Hosokawa,
2t^{-2}+t^{-1}-7+t+2t^2 $2t^{-2}+t^{-1}-7+t+2t^2$ 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.
