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 $(p\ge 2,q)$-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 $t(K^{\star })\le t(K)+1$, Example 1.1 in Section 1 shows that in some case, $t(K^{\star })\le t(K)$. 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)ifeither$K$admitsahighdistanceHeegaardsplittingor$p/q$ 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.