The trunk number of satellite knots and Thurston norm

Abstract

Assume $J\subset {\mathbb{R}}^3$ is a non-trivial knot, and assume $\hat{k}\subset S^1\times D^2$ is a satellite pattern. Let $N$ be the generalized Thurston norm of the homology class of the meridian disk in $S^1\times D^2$ with respect to $\hat{k}$. Let $K$ be the satellite knot of $J$ with pattern $\hat{k}$. We show that the trunk number of $K$ is strictly greater than $N$ times the trunk number of $J$.