  Using an involved study of the Jones polynomial, we determine, as our main
result, the crossing numbers of (prime) amphicheiral knots. As further
applications, we show that several classes of links, including semiadequate
links and Whitehead doubles of semiadequate knots, have non-trivial Jones
polynomial. We also prove that there are infinitely many positive knots with no
positive minimal crossing diagrams. Some relations to the twist number of a
link, Mahler measure and the hyperbolic volume are given, for example explicit
upper bounds on the volume for Montesinos and 3-braid links in terms of their
Jones polynomial.
