论文标题
关于电缆结的隧道数量及其同伴
On tunnel numbers of a cable knot and its companion
论文作者
论文摘要
令$ k $为$ s^{3} $和$ t（k）$的隧道号码的非平凡结。对于任何$（p \ geq 2，q）$ - 在$ s^{3} $的$ k $ in $ k $ in $ s^{3} $中的斜坡上的斜率，由$ k^{\ star} $表示，这是$ s^{3} $的非平凡的电缆结。尽管$ t（k^{\ star}）\ leq t（k）+1 $，但第1节中的示例1.1表明，在某些情况下，$ t（k^{\ star}）\ leq t（k）$。因此，很有趣的是知道$ t（k^{\ star}）= t（k）+1 $。 使用了一些组合技术后，我们证明（1）对于任何非平凡的电缆结$ k^{\ star} $及其伴侣$ k $，$ t（k^{\ star}）\ geq t（k）$; （2）如果$ k $承认高距离heegaard拆分或$ p/q $远离福利图中的固定子集，则$ 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侣，以使其隧道数量之间的差异很大。
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.
