The genus 1 bridge number of satellite knots

IF 1.2 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-08-11 DOI:10.1112/jlms.70260

Scott A. Taylor, Maggy Tomova

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引用次数: 0

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 $ω$ with respect to $K$ . When $K$ is not a torus knot, we show that $b_{1} (T) ⩾ ω b_{1} (K)$ . 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 $b_{0} (T) ⩾ ω b_{0} (K)$ . We also prove versions of the theorem applicable to when $T$ is a “lensed satellite” and when there is a torus separating components of $T$ .

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属1桥数卫星节

设T $T$为3流形M $M$中的卫星结、链路或空间图，该3流形M 是s3 $S^3$或透镜空间。设b0 $\mathfrak {b}_0$和b1 $\mathfrak {b}_1$分别表示0和1属桥号。假设T $T$有一个伴结K $K$（不一定是解结）和相对于K $K$的包裹数ω $\omega$。当K $K$不是环面结时，我们显示b1 (T)小于ω b1 (K) $\mathfrak {b}_1(T)\geqslant \omega \mathfrak {b}_1(K)$。如果K $K$是环面结，有一些已知的反例。一路走来，我们推广并给出舒伯特结果的新证明b 0 (T)小于ω b 0 (K)$\mathfrak {b}_0(T) \geqslant \omega \mathfrak {b}_0(K)$。我们还证明了当T $T$是一个“透镜卫星”以及当有一个环面分离T $T$的组件时适用的定理版本。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.