On the growth rate of tunnel number of knots

2004/02/03 by Tsuyoshi Kobayashi, Kobayashi, Tsuyoshi, Yo'av Rieck +2 · 1 citation
Computer Science · Mathematics · #57M99 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:57M99 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0402025

19 pages, 8 figures

arxiv created 2004/02/03 · openalex publication_date 2004/02/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a knot K in a closed orientable manifold M we define the growth rate of the tunnel number of K to be gr_t(K) = \limsup_{n → ∞} (t(nK) - n t(K))/(n-1). As our main result we prove that the Heegaard genus of M is strictly less than the Heegaard genus of the knot exterior if and only if the growth rate is less than 1. In particular this shows that a non-trivial knot in S³ is never asymptotically super additive. The main result gives conditions that imply falsehood of Morimoto's Conjecture.

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• Heegaard genus of the connected sum of m-small knots

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