Knots and shellable cell partitionings of \(S^ 3\)
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.
scientific article; zbMATH DE number 599236