Tunnel number one knots, \(m\)-small knots and the Morimoto conjecture
type: Item
instance of: scholarly article
zbMATH Open document ID: 1282.57015
DOI: 10.1090/S0002-9939-2013-11700-4
author: Xunbo Yin
author: Guoqiu Yang
author: Fengchun Lei
published in: Proceedings of the American Mathematical Society
publication date: 2013-10-23T00:00:00Z
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6218167
zbMATH Keywords: tunnel number
zbMATH Keywords: \(m\)--small
zbMATH Keywords: Morimoto conjecture
Wikidata QID: Q123367162
MaRDI profile type: Publication
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title: Tunnel number one knots, \(m\)-small knots and the Morimoto conjecture
review text: The tunnel number \(t(K)\) of a knot \(K\) in the 3--sphere \(S^3\) is the least number of arcs that can be attached to \(K\) to produce a graph whose complement in \(S^3\) is a handlebody. Given two knots \(K_1\) and \(K_2\) in \(S^3\), the tunnel number of their connected sum satisfies \(t(K_1 \# K_2) \leq t(K_1) + t(K_2) + 1\). The original Morimoto conjecture asserted that equality held if and only if neither \(K_1\) nor \(K_2\) was \(\mu\)--primitive, meaning that their knot complements had a minimal primitive Heegard splitting. Counterexamples were found and the conjecture was revised to apply only to prime knots. (See [\textit{Y. Moriah}, Proceedings of the Technion workshop on Heegaard splittings, Haifa, Israel, summer 2005. Coventry: Geometry \& Topology Publications. Geometry and Topology Monographs 12, 191--232 (2007; Zbl 1138.57008)]) In the paper under review, the authors prove the conjecture when each of \(K_1\) and \(K_2\) either have tunnel number one or are \(m\)--small, meaning that their knot complement contains no meridional essential surface.
reviewed by: James Hebda
description: scientific article; zbMATH DE number 6218167