Properties of minimal charts and their applications. II (Q1026929)

Charts were introduced by \textit{S. Kamada} [J. Knot Theory Ramifications 1, No.~2, 137--160 (1992; Zbl 0763.57013)] as a tool for representing surface braids. A chart is an oriented planar graph \(\Gamma\) with edges labeled by natural numbers, satisfying certain conditions. In particular, the vertices of \(\Gamma\) are allowed to have degree 1 (black vertices), degree 4 (crossings), and degree 6 (white vertices). Moreover, the edges converging at any white vertex are alternately labeled by \(i\) and \(i+1\) for some \(i \geq 1\), and we call that vertex an \(i\)-vertex. Two \(n\)-charts represent the same surface braid if and only if they are equivalent up to certain local modifications called \(C\)-moves. The complexity of a chart \(\Gamma\) is the pair \((w(\Gamma),-f(\Gamma))\), where \(w(\Gamma)\) is the number of white vertices and \(f(\Gamma)\) is the number of edges joining two black vertices. Then \(\Gamma\) is called a minimal chart if it has minimal complexity among the charts \(C\)-move equivalent to it with respect to the lexicographic order. It is known that there is no minimal chart \(\Gamma\) with \(w(\Gamma) = 1,2,3,5\), while there exists such a minimal chart with \(w(\Gamma) = 4,6\). In this paper, the author continue their investigation on minimal charts, started with [J. Math. Sci., Tokyo 14, No.~1, 69--97 (2007; Zbl 1135.57012)] and aimed to prove that there are no minimal charts with \(w(\Gamma) = 7\). The main result here is that if \(\Gamma\) is a minimal chart with \(w(\Gamma) = 7\) then the sequence \((n_1, \dots, n_k)\), with \(n_i\) the number of white \(i\)-vertices of \(\Gamma\), is one of the following, up to translation or reversion of the labeling: \((7)\), \((5,2)\), \((4,3)\), \((3,2,2)\), \((2,3,2)\).