Width of a satellite knot and its companion

Abstract

In this paper, we give a proof of a conjecture which says that $w\left(K\right)\ge n^{2}w\left(J\right)$, where $w\left(\cdot \right)$ is the width of a knot, $K$ is a satellite knot with $J$ as its companion, and $n$ is the winding number of the pattern. We also show that equality holds if $K$ is a satellite knot with braid pattern.