Destabilizing Heegaard splittings of knot exteriors
type: Item
instance of: scholarly article
title: Destabilizing Heegaard splittings of knot exteriors
zbMATH Open document ID: 1186.57016
author: Tsuyoshi Kobayashi
author: Toshio Saito
published in: Topology and its Applications
publication date: 2009-12-03T00:00:00Z
review text: It is known that the connected sum of minimal genus Heegaard splittings yields a minimal genus Heegaard splitting. This follows from \textit{W. Haken's} lemma [Stud. in Math. 5 (Studies Modern Topol.), 39--98 (1968; Zbl 0194.24902)]. On the other hand, Gordon asked the following: Can the connected sum of unstabilized Heegaard splittings yield a stabilized Heegaard splitting? (See Problem 3.9 in [\textit{R. Kirby}, Problems in low-dimensional topology, in: Geometric topology. 1993 Georgia international topology conference,1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.2), 35--473 (1997; Zbl 0888.57014)].) In this nice paper, the authors give a necessary and sufficient condition for Heegaard splittings of knot exteriors to admit destabilizations. Then they examine the pair of knots \(K_1\) and \(K_2\), which was introduced by \textit{K. Morimoto} [Topology Appl. 64, No.~2, 165--176 (1995; Zbl 0827.57002)], as an example giving degeneration of tunnel number under connected sum. For this pair, the authors show that the Heegaard splitting of the exterior of \(K_1\# K_2\) derived from certain minimal unknotting tunnel systems of \(K_1\) and \(K_2\) is stabilized.
reviewed by: Alberto Cavicchioli
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5641763
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: unknotting tunnel system
zbMATH Keywords: destabilization
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.059
OpenAlex ID: W1964253039
cites work: Connected sums of unstabilized Heegaard splittings are unstabilized
cites work: Reducing Heegaard splittings
cites work: Q5589863
cites work: Q4382039
cites work: Separating incompressible surfaces and stabilizations of Heegaard splittings
cites work: On the growth rate of the tunnel number of knots
cites work: Knots with \(g(E(K))=2\) and \(g(E(K\#K\#K))=6\) and Morimoto's conjecture
cites work: Knot exteriors with additive Heegaard genus and Morimoto's conjecture
cites work: An algorithm to determine the Heegaard genus of simple 3-manifolds with nonempty boundary
cites work: Characterization of tunnel number two knots which have the property ``\(2+1= 2\)''
cites work: Q4234977
cites work: A proof of the Gordon conjecture
cites work: The Classification of Heegaard Splittings for (Compact Orient Able Surface) × S1
cites work: Q5457876
DOI: 10.1016/J.TOPOL.2009.04.059
description: scientific article; zbMATH DE number 5641763