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.

11 pages, 5 Figures. To appear in The Bulletin of London Math Society. arXiv admin note: text overlap with arXiv:2108.12391