Knots and shellable cell partitionings of \(S^ 3\)
type: Item
instance of: scholarly article
title: Knots and shellable cell partitionings of \(S^ 3\)
zbMATH Open document ID: 0832.57014
author: Steve Armentrout
published in: Illinois Journal of Mathematics
publication date: 1996-03-07T00:00:00Z
review text: A shelling of a cell-decomposed \(d\)-sphere is a numbering \(h_1, \dots, h_n\) of the cells such that, after removing \(h_1, \dots, h_i\) for \(i\geq 1\), one always gets a \(d\)-ball. Unshellable triangulations of spheres have been obtained by \textit{W. B. R. Lickorish} [Eur. J. Comb. 12, No. 6, 527-530 (1991; Zbl 0746.57007)]. The present paper gives a systematic way how to construct non-shellable cell-decompositions by means of knots. Roughly speaking, a knot is called compatible with the decomposition if it meets each cell either in a single arc or not at all. Theorem 1 states that if the number \(\pi\) of those arcs is strictly smaller than twice the bridge number \(b\) of the knot, then the cell decomposition is not shellable. This inequality is sharp in that for every knot there are shellable cell-decompositions satisfying \(\pi= 2b\). Explicit examples are given for non-shellable cell-decompositions. One of them is based on the trefoil knot.
Mathematics Subject Classification ID: 57Q99
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 599236
zbMATH Keywords: shelling
zbMATH Keywords: \(d\)-sphere
zbMATH Keywords: non-shellable cell-decompositions
zbMATH Keywords: knots
zbMATH Keywords: bridge number
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 599236