Common stabilizations of Heegaard splittings of link exteriors
type: Item
instance of: scholarly article
title: Common stabilizations of Heegaard splittings of link exteriors
zbMATH Open document ID: 1068.57016
author: Jung Hoon Lee
author: Gyo Taek Jin
published in: Topology and its Applications
publication date: 2005-06-01T00:00:00Z
review text: A \textit{compression body} \(H\) is a 3-manifold obtained from a compact closed surface \(S\) by attaching 2-handles to \(S\times [0,1]\) on \(S\times \{1\}\) and capping off any resulting 2-sphere boundary component with 3-handles. The surface \(S\times \{0\}\) is denoted by \(\partial_{+}H\) and \(\partial H-\partial_{+}H\) is denoted by \(\partial_{-}H\). If a compact 3-manifold \(M\) is the union of two compression bodies \(H_1\) and \(H_2\) along their boundary \(S=\partial_{+}H_1=\partial_{+}H_2\), we call the decomposition \(M=H_1\cup_{S}H_2\) a \textit{Heegaard splitting } of \(M\). The genus of \(S\) is called the genus of the Heegaard splitting of \(H_1\cup_{S}H_2\). Two Heegaard splittings \(H_1\cup_S H_2\) and \(H'_1\cup_{S'}H'_2\) of \(M\) are \textit{isotopic} if there is an ambient isotopy \(h_t\,\,\,(0\leq t \leq 1)\) of \(M\) such that \(h_1(H_1)=H'_1\). Let \(V_1\cup_{T}V_2\) be the genus one Heegaard splitting of the 3-sphere \(S^3\). Note that genus \(g\) Heegaard splittings of \(S^3\) are mutually isotopic for each \(g\) by \textit{F. Waldhausen} [Topology 7, 195--203 (1968; Zbl 0157.54501)]. We say \(H'_{1}\cup_{S'}H'_{2}\) is obtained from \(H_1\cup_{S}H_2\) by a \textit{stabilization} if \(H'_{1}\cup_{S'}H'_{2}\) is a connected sum of \(H_1\cup_{S}H_2\) and \(V_1\cup_{T}V_2\). \textit{M. Reidemeister} [Abhandlungen Hamburg 10, 109--118 (1934; JFM 60.0514.01)] and \textit{J. Singer} [Trans. Am. Math. Soc. 35, 88--111 (1933; Zbl 0006.18501)] showed that any two Heegaard splittings of \(M\) become isotopic by a sequence of stabilizations, and then the next conjecture was presented by \textit{M. Scharlemann} in [Handbook of geometric topology, Amsterdam: Elsevier, 921--953 (2002; Zbl 0985.57005)]: Conjecture. Suppose \(H_1\cup_{S}H_2\) and \(H'_1\cup_{S'}H'_2\) are Heegaard splittings of the same 3-manifold: genus \(g\leq g'\), respectively. Then the splittings obtained by one stabilization of \(H'_1\cup_{S'}H'_2\) and \(g'-g+1\) stabilizations …
reviewed by: Hiroshi Goda
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 2173257
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: stabilization
zbMATH Keywords: tunnel number
zbMATH Keywords: link exterior
MaRDI profile type: Publication
full work available at URL: j.topol.2004.11.005
OpenAlex ID: W1986396010
cites work: An alternative proof that 3-manifolds can be triangulated
cites work: On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces
cites work: Q4328307
cites work: Q4486720
cites work: Heegaard splittings of exteriors of two bridge knots
cites work: TUNNELS OF 2-BRIDGE LINKS
cites work: Affine structures in 3-manifolds. V: The triangulation theorem and Hauptvermutung
cites work: On unknotting tunnels for knots
cites work: Heegaard splittings of compact 3-manifolds
cites work: Heegaard splittings of \((\text{surface})\times I\) are standard
DOI: 10.1016/J.TOPOL.2004.11.005
description: scientific article; zbMATH DE number 2173257