A note on tunnel number of composite knots
type: Item
instance of: scholarly article
title: A note on tunnel number of composite knots
zbMATH Open document ID: 1228.57003
author: Xutao Gao
author: Qilong Guo
author: Ruifeng Qiu
published in: Topology and its Applications
publication date: 2011-10-10T00:00:00Z
review text: For a knot \(K\) in the \(3\)-sphere, its tunnel number \(t(K)\) is essentially the Heegaard genus of the knot exterior. There are various results about the behavior of tunnel number under the connected sum operation between knots. It is easy to see that \(t(K_1\sharp K_2)\leq t(K_1)+t(K_2)+1\) for two knots \(K_1\) and \(K_2\). In fact, \textit{T. Kobayashi} [J. Knot Theory Ramifications 3, No.~2, 179--186 (1994; Zbl 0818.57006)] proved that the gap between \(t(K_1\sharp K_2)\) and \(t(K_1)+t(K_2)\) can be arbitrarily large. On the other hand, the equality \(t(K_1\sharp K_2)=t(K_1)+t(K_2)+1\) holds for various cases. For example, \textit{F. Li, G. Yang} and \textit{F. Lei} [Topology Appl. 157, No.~7, 1188--1194 (2010; Zbl 1189.57020)] showed that if the distance \(d(K_i)\) is at least \(2t(K_i)+5\) for \(i=1,2\), then the above equality holds. Here, the distance \(d(K)\) is defined to be the minimal distance among all the minimal Heegaard splittings of the knot exterior of a knot \(K\). The main result of the paper under review is to give a new sufficient condition for the equality. More precisely, if \(d(K_1)\geq 3\) and \(d(K_2)\geq 2(t(K_1)+t(K_2)+2)\), then \(t(K_1\sharp K_2)=t(K_1)+t(K_2)+1\). Editor's remark: the erratum does not concern the contents of the present article but only the first and second author's affiliations.
reviewed by: Masakazu Teragaito
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 5956074
zbMATH Keywords: tunnel number
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: high distance
MaRDI profile type: Publication
full work available at URL: j.topol.2011.07.020
OpenAlex ID: W2034643573
cites work: Reducing Heegaard splittings
cites work: HEEGAARD GENERA OF ANNULAR 3-MANIFOLDS
cites work: Heegaard splittings of Haken manifolds have bounded distance.
cites work: 3-manifolds as viewed from the curve complex
cites work: A CONSTRUCTION OF ARBITRARILY HIGH DEGENERATION OF TUNNEL NUMBERS OF KNOTS UNDER CONNECTED SUM
cites work: Heegaard genera of high distance are additive under annulus sum
cites work: High distance knots
cites work: There are Knots whose Tunnel Numbers go down under Connected Sum
cites work: On the super additivity of tunnel number of knots
cites work: The Heegaard genera of surface sums
cites work: Proximity in the curve complex: boundary reduction and bicompressible surfaces
cites work: Q4307578
cites work: Alternate Heegaard genus bounds distance
cites work: Additivity of tunnel number for small knots
DOI: 10.1016/J.TOPOL.2011.07.020
description: scientific article; zbMATH DE number 5956074