The Jones polynomial of parallels and applications to crossing number (Q756780)

We study the Jones polynomial of the parallels of a link or knot. From the extremal exponents occurring we derive lower bounds 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.