A generalization of the rectangle condition
type: Item
instance of: scholarly article
title: A generalization of the rectangle condition
zbMATH Open document ID: 1395.57017
published in: Topology and its Applications
publication date: 2018-06-07T00:00:00Z
review text: Let \(M\) be a closed oriented \(3\)-manifold. If there is a closed surface \(S\) which cuts \(M\) into two handlebodies \(V\) and \(W\) with \(S=\partial_{+} V=\partial_{+} W\), then we say \(M\) has a Heegaard splitting, denoted by \(M=V\cup_{S} W\). \textit{A. Casson} and \textit{C. Gordon} [``Manifolds with irreducible Heegaard splittings of arbitrary large genus'', Unpublished] introduced the rectangle condition on Heegaard surfaces to show strong irreducibility of Heegaard splittings. In the paper under review, the author introduces the connecting rectangle condition and a general version of the rectangle condition to check whether or not \(M=V\cup_{S} W\) is strongly irreducible. Also, the author introduces the connecting rectangle condition on the \(n\)-bridge sphere for an \(n\)-bridge decomposition of a knot and checks whether or not the Hempel distance of an \(n\)-bridge decomposition is greater than or equal to two. Finally, the author gives an interesting example of an \(n\)-bridge decomposition of links which satisfy the connecting rectangle condition.
reviewed by: Kun Du
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 6882111
zbMATH Keywords: connecting rectangle condition
zbMATH Keywords: various pants decomposition
zbMATH Keywords: train track
author: Bo-hyun Kwon
MaRDI profile type: Publication
full work available at URL: j.topol.2018.05.005
OpenAlex ID: W2804034033
cites work: Distance and bridge position
cites work: Reducing Heegaard splittings
cites work: 3-manifolds as viewed from the curve complex
cites work: An algorithm to classify rational 3-tangles
cites work: Uniqueness of bridge surfaces for 2-bridge knots
cites work: Bridge decompositions with distances at least two
Wikidata QID: Q129789449
DOI: 10.1016/J.TOPOL.2018.05.005
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description: scientific article; zbMATH DE number 6882111