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 ≥ 2 , q ) -slope in the torus boundary of a closed regular neighborhood of K in S 3 , denoted by K ⋆ , it is a nontrivial cable knot in S 3 . 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.