2020/02/18 by Junhua Wang, Wang, Junhua, Yanqing Zou +1
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2002.07326
openalex publication_date 2020/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.