Crossing numbers of cable knots

Abstract

We use the degree of the colored Jones knot polynomials to show that the crossing number of a $(p,q)$-cable of an adequate knot with crossing number $c$ is larger than $q^{2}c$. As an application, we determine the crossing number of 2-cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a 2-cable of an adequate knot.