Morimoto's Conjecture for m-small knots

2002/12/27 by Tsuyoshi Kobayashi, Kobayashi, Tsuyoshi, Yo'av Rieck +2
Mathematics · #57M99 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M99

paper · pdf · doi:10.48550/arxiv.math/0212349

17 pages; to appear in the proceedings of the conference "Musubime no topology (Topology of knots) V" held at Waseda University, 16-19 December, 2002

arxiv created 2002/12/27 · openalex publication_date 2002/12/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be the exterior of connected sum of knots and X_i the exteriors of the individual knots. In \citemorimoto1 Morimoto conjectured (originally for n=2) that g(X) < σ_i=1ⁿ g(X_i) if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of the knots. For m-small knots we prove this conjecture and bound the possible degeneration of the Heegaard genus (this bound was previously achieved by Morimoto under a weak assumption \citemorimoto2): σ_i=1ⁿ g(X_i) - (n-1) ≤ g(X) ≤ σ_i=1ⁿ g(X_i).

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