Let Jk(*) = nrtr + • ■ • + asta, r > s, be the Jones polynomial of a knot if in S3. For an alternating knot, it is proved that r — s is bounded by the number of double points in any alternating projection of K. This upper bound is attained by many alternating knots, including 2-bridge knots, and therefore, for these knots, r — s gives the minimum number of double points among all alternating projections of K. If K is a special alternating knot, it is also proved that a3 = 1 and s is equal to the genus of K. Similar results hold for links. 1. Introduction. Let K be an oriented knot or link in S3 and let Jx(t) be the polynomial defined by V. Jones (5) which is now called the Jones polynomial of K. Jk (t) is an invariant of a knot or link type. It is not clear, however, to what extent Jk (t) is related to known algebraic or topological invariants in knot theory. Although it is shown (10) that Jk (t) determines the Arf invariant of a knot or a link, it is also known that Jk(i) does not determine the genus or the signature of K(8}. In this paper, we prove that for an alternating knot, Jk(t) provides some infor- mation that has never been obtained from other algebraic invariants. To be more precise, let K be an alternating knot in S3 and K an alternating projection of K. Let Jk(t) be the Jones polynomial of K.1 Write Jk(i) — artr + • • • + asts, where r > s and ar ^ 0 / as. Let h(K) be the number of double points in K. Then we can prove