Properties of minimal charts and their applications. IX: Charts of type \((4, 3)\) (Q6041358)

In this paper, the main result is that there exists no minimal chart of type \((4,3)\); this result, together with other published or future results by the authors, shall show that there is no minimal chart with exactly seven white vertices. \newline For a chart \(\Gamma\) with at least one white vertex, and integers \(n_1, n_2, \ldots, n_k\), \(\Gamma\) is of type \((n_1, n_2, \ldots, n_k)\) if there exists a label \(m\) of \(\Gamma\) such that \(\Gamma\) has exactly \(\sum_{i=1}^k n_i\) white vertices and there are exactly \(n_i\) white vertices in \(\Gamma_{m+i-1} \cap \Gamma_{m+i}\) \((i=1, \ldots,k)\). Here, \(\Gamma_i\) is the union of all the edges of label \(i\). And the authors showed in [Hiroshima Math. J. 39, No. 1, 1--35 (2009; Zbl 1194.57030)] that if there exists a minimal chart with exactly seven white vertices, then the chart is of type \((7)\), \((5,2)\), \((4,3)\), \((3,2,2)\) or \((2,3,2)\). \newline In the proof, the authors investigate forms of \(k\)-angled disks with or without feelers for some cases. Here, a disk \(D\) with \(\partial D \subset \Gamma_m\) is called a \(k\)-angled disk of \(\Gamma_m\) if \(\partial D\) contains exactly \(k\) white vertices. And an edge of label \(m\) is called a feeler of a \(k\)-angled disk \(D\) of \(\Gamma_m\) if the edge intersects \(N-\partial D\), where \(N \subset D\) is a regular neighborhood of \(\partial D\). \newline We review the main terms. A chart is an oriented labeled graph in a disk satisfying certain conditions, equipped with three types of vertices: vertices of degree 1 called black vertices, and vertices of degree 4 called crossings, and vertices of degree 6 called white vertices. An edge in a chart is called a free edge if it has two black vertices at its endpoints. A chart presents an oriented surface link in \(\mathbb{R}^4\), which is a closed surface embedded in \(\mathbb{R}^4\). A C-move is a local modification between two charts of the same degree, and a C-move between two charts induces an ambient isotopy between oriented surface links presented by the charts. For a chart \(\Gamma\), the pair \((w(\Gamma), -f(\Gamma))\) is called the complexity of \(\Gamma\), where \(w(\Gamma)\) and \(f(\Gamma)\) are the number of white vertices and the number of free edges, respectively, and a chart \(\Gamma\) is minimal if its complexity is minimal with respect to the lexicographic order of pairs of integers, among charts which are related to \(\Gamma\) by C-moves.