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On tunnel numbers of a cable knot and its companion

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.