The Jones Polynomial of an Alternating Link

Abstract

This chapter contains some of the most impressive applications of the Jones polynomial. They give solutions to two problems encountered by P. G. Tait in the nineteenth century. It is shown that an alternating knot diagram, when “reduced” in a rather elementary way, has the minimal number of crossings and that its writhe is an invariant of the knot. The crossing number of some other types of knot is also determined.