Properties of minimal charts and their applications. III (Q632487)

The notion of a chart was introduced by \textit{S. Kamada} [J. Knot Theory Ramifications 1, No. 2, 137--160 (1992; Zbl 0763.57013)] as a method to present a surface braid in 4-space. A chart is an oriented and labeled graph in a disk satisfying certain conditions, with vertices of degree 1 (black vertices), degree 4 (crossings) and degree 6 (white vertices). For two charts, their presenting surface braids are equivalent if and only if the charts are C-move equivalent i.e. they are related by local modifications called C-moves, cf. \textit{S. Kamada} [J. Knot Theory Ramifications 5, No. 4, 517--529 (1996; Zbl 0889.57011)]. The complexity of a chart \(\Gamma\) is the pair \((w(\Gamma), -f(\Gamma))\), where \(w(\Gamma)\) is the number of white vertices of \(\Gamma\) and \(f(\Gamma)\) is the number of edges of \(\Gamma\) whose endpoints are black vertices. A minimal chart is a chart whose complexity is minimal among the charts in its C-move equivalence class, with respect to the lexicographic order. This paper is a continuation of the investigation on minimal charts by the authors [Part I, J. Math. Sci., Tokyo 14, No.~1, 69--97 (2007; Zbl 1135.57012)] and [Part II, Hiroshima Math. J. 39, No.~1, 1--35 (2009; Zbl 1194.57030)] with the aim of showing that there is no minimal chart with exactly seven white vertices. The authors investigate a disk called a \(k\)-angled disk for a minimal chart. Let \(\Gamma\) be a minimal chart. A \(k\)-angled disk is a disk \(D\) such that \(\partial D\) consists of \(k\) edges of \(\Gamma\) with the same label. The local complexity with respect to \(D\) is the pair \((w(D), c(D))\), where \(w(D)\) is the number of white vertices of \(\Gamma\) in \(\mathrm{Int}(D)\) and \(c(D)\) is the number of crossings of \(\Gamma\) on \(\partial D\). The notion of a chart locally minimal with respect to \(D\) is defined by the similar way to the notion of a minimal chart, by using the local complexity. The main result in this paper is that for a minimal chart \(\Gamma\) and a specific 2-angled (respectively 3-angled) disk \(D\) such that \(\Gamma\) is locally minimal with respect to \(D\), if there is at most one white vertex in \(\mathrm{Int}(D)\), then a regular neighborhood of \(D\) contains an element of five (respectively eight) graphs called pseudo charts as shown in the paper, up to reflection or orientation-reversal.