2024/01/03 by Taylor, Scott A., Tomova, Maggy · 1 citation
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2401.01834
Let T be a satellite knot, link, or spatial graph in a 3-manifold M that is either S3 or a lens space. Let \mathfrakb0 and \mathfrakb1 denote genus 0 and genus 1 bridge number, respectively. Suppose that T has a companion knot K (necessarily not the unknot) and wrapping number ω with respect to K. When K is not a torus knot, we show that \mathfrakb1(T)≥ ω\mathfrakb1(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 \mathfrakb0(T) ≥ ω\mathfrakb0(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.