Tunnel number, connected sum and meridional essential surfaces
type: Item
instance of: scholarly article
title: Tunnel number, connected sum and meridional essential surfaces
zbMATH Open document ID: 0960.57006
DOI: 10.1016/S0040-9383(98)00070-6
author: Kanji Morimoto
published in: Topology
publication date: 2001-05-10T00:00:00Z
review text: Let \(K_1,\ldots,K_n\) be knots in \(S^3\) and \(t(K_i)\), \(i=1,\ldots,n\) the tunnel numbers of them. \textit{K. Morimoto} and \textit{J. Schultens} [Proc. Am. Math. Soc. 128, No. 1, 269-278 (2000; Zbl 0928.57003)] showed that if none of the knot exteriors \(E(K_1),\ldots,E(K_n)\) contains closed essential surfaces, then \(t(K_1\#\ldots\#K_n)\geq t(K_1)+\ldots+t(K_n)\). This result is here generalized to show that, for given knots \(K_1,\ldots,K_n\) in orientable closed \(3\)-manifolds \(M_1,\ldots,M_n\) respectively, if none of those \(3\)-manifolds has lens space summands and none of those knot exteriors contains meridional essential surfaces, then the inequality above holds. The proof depends on the concept of strongly irreducible Heegaard splittings which has been developed by \textit{M. Scharlemann} and \textit{A. Thompson} [Contemp. Math. 164, 231-238 (1994; Zbl 0818.57013)] together with a concept of an amalgamation of Heegaard splittings due to \textit{J. Schultens} [Proc. Lond. Math. Soc., III. Ser. 67, No. 2, 425-448 (1993; Zbl 0789.57012)].
reviewed by: Leila Ben Abdelghani
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1436046
zbMATH Keywords: strongly irreducible Heegaard splittings
zbMATH Keywords: amalgamation of Heegaard splittings
MaRDI profile type: Publication
Wikidata QID: Q127006459
description: scientific article; zbMATH DE number 1436046