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On tunnel numbers of a cable knot and its companion
Topology and its Applications ( IF 0.6 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.topol.2020.107319
Junhua Wang , Yanqing Zou

Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $ K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some case, $ t(K^{\star})\leq t(K)$. So it is interesting to know when $t(K^{\star})= t(K)+1$. After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot $K^{\star}$ and its companion $K$, $t(K^{\star})\geq t(K)$; (2) if either $K$ admits a high distance Heegaard splitting or $p/q$ is far away from a fixed subset in the Farey graph, then $t(K^{\star})= t(K)+1$. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

中文翻译:

关于电缆结及其伴体的隧道编号

令 $K$ 是 $S^{3}$ 中的一个重要结,$t(K)$ 是其隧道编号。对于 $S^{3}$ 中 $K$ 的封闭正则邻域的环面边界中的任何 $(p\geq 2,q)$-斜率,用 $K^{\star}$ 表示,它是$S^{3}$ 中的非平凡电缆结。尽管 $t(K^{\star})\leq t(K)+1$,第 1 节中的示例 1.1 表明,在某些情况下,$ t(K^{\star})\leq t(K)$。所以知道什么时候 $t(K^{\star})= t(K)+1$ 很有趣。在使用一些组合技术之后,我们证明了 (1) 对于任何非平凡的缆绳结 $K^{\star}$ 及其同伴 $K$,$t(K^{\star})\geq t(K)$;(2) 如果 $K$ 承认高距离 Heegaard 分裂或 $p/q$ 远离 Farey 图中的固定子集,则 $t(K^{\star})= t(K)+1 $. 使用第二个结论,我们构建了一个卫星结及其伴星,使得它们的隧道编号之间的差异是任意大的。
更新日期:2020-08-01
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