Jones Polynomials of Alternating Links

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.