On tunnel numbers of a cable knot and its companion

Let K be a nontrivial knot in S³ and t(K) its tunnel number. For any (p≥2,q)-slope in the torus boundary of a closed regular neighborhood of K in S³, denoted by K^⋆, it is a nontrivial cable knot in S³. Though t(K^⋆)≤t(K)+1, Example 1.1 in Section 1 shows that in some case, t(K^⋆)≤t(K). So it is interesting to know when t(K^⋆)=t(K)+1. After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot K^⋆ and its companion K, t(K^⋆)≥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^⋆)=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.