Jones Polynomials of Alternating Links

Abstract

Let ${J_K}(t) = {a_r}{t^r} + \cdots + {a_s}{t^s},r > s$, be the Jones polynomial of a knot $K$ in ${S^3}$. 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 ${a_s} = 1$ and $s$ is equal to the genus of $K$. Similar results hold for links.