An official website of the United States government
Abstract 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 $$\b_1(T)\geq \omega \b_1(K)$$. There are previously known counter-examples if $$K$$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $$\b_0(T) \geq \omega \b_0(K)$$. We also prove versions of the theorem applicable to when $$T$$ is a "lensed satellite" and when there is a torus separating components of $$T$$.
more »
« less
Free, publicly-accessible full text available August 1, 2026
