A volumish theorem for the Jones polynomial of alternating knots

The Volume Conjecture claims that the hyperbolic volume of a knot is determined by the colored Jones polynomial. Here we prove a "Volumish Theorem" for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the volume and certain sums of coefficients of the Jones polynomial is bounded from above and from below by constants. Furthermore, we give experimental data on the relation of the growths of the hyperbolic volume and the coefficients of the Jones polynomial, both for alternating and nonalternating knots.