The Jones polynomial of parallels and applications to crossing number

In this paper we study the Jones polynomial of the parallels of a link or knot.From the extremal exponents occurring we derive lower hounds on the crossing number of the knot, reproducing in particular a number of results of Thistlethwaite.We apply these techniques to give lower bound on the crossing number of some simple satellites of adequate and semi-adequate knots (including cable satellites) that are usually quadratic in the degree of the satellite.PROPOSITION 1.(a) If D be a link diagram with n crossings then M(D) -n -2-D\ + 2 with equality if D is -adequate.Proof, (a) Clearly M(D+) = n+2+D\-2.Suppose s is any other state.Then there is a sequence of states 5+ = SQ , s\, ... , s^ = s, such