Crossing numbers of cable knots

TL;DR

This paper establishes a lower bound for the crossing number of cable knots using colored Jones polynomials and determines exact crossing numbers for certain cable knots and their connected sums.

Contribution

It introduces a method to estimate crossing numbers of cable knots via colored Jones polynomials and provides explicit crossing number calculations for specific cable and connected sum cases.

Findings

Crossing number of a (p,q)-cable exceeds q^2 times the original crossing number.

Exact crossing numbers are determined for 2-cables of adequate knots.

Crossing numbers of connected sums involving 2-cables are explicitly calculated.

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.