2020/07/29 by Tao Li, Li, Tao
Mathematics · Medicine · #57K10 #57K30 #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2007.14534
openalex publication_date 2020/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M=W∪T V be an amalgamation of two compact 3-manifolds along a torus, where W is the exterior of a knot in a homology sphere. Let N be the manifold obtained by replacing W with a solid torus such that the boundary of a Seifert surface in W is a meridian of the solid torus. This means that there is a degree-one map f\colon M→ N, pinching W into a solid torus while fixing V. We prove that g(M)≥ g(N), where g(M) denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.