The genus 1 bridge number of satellite knots

Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $b_0$ and $b_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion knot $K$ (necessarily not the unknot) and wrapping number $\omega$ with respect to $K$. When $K$ is not a torus knot, we show that $b_1\left(T\right)⩾\omega b_1\left(K\right)$. There are previously known counterexamples if $K$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $b_0\left(T\right)⩾\omega b_0\left(K\right)$. We also prove versions of the theorem applicable to when $T$ is a “lensed satellite” and when there is a torus separating components of $T$.