On charts with two crossings. I: There exist no NS-tangles in a minimal chart (Q2902433)

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; see \textit{S. Kamada} [J. Knot Theory Ramifications 5, No. 4, 517--529 (1996; Zbl 0889.57011)]. The extended complexity of a chart \(\Gamma\) is the triple \((w(\Gamma), -f(\Gamma), -b(\Gamma))\), where \(w(\Gamma)\) is the number of white vertices in \(\Gamma\), \(f(\Gamma)\) is the number of the edges in \(\Gamma\) whose endpoints are black vertices, and \(b(\Gamma)\) is the number of particular areas called bigons in \(\Gamma\). A \(k\)-minimal chart is a chart whose extended complexity is minimal among its C-move equivalent charts with at most \(k\) crossings, with respect to the lexicographic order.NEWLINENEWLINEIn this paper, the authors establish methods to count crossings and terminal edges in \(k\)-minimal charts (Theorems 3.5, 4.8, 5.4). Here, a terminal edge is an edge whose endpoints are a white vertex and a black vertex. The title is given from Theorem 3.5 which shows that there does not exist any partial graph called NS-tangle in a \(k\)-minimal chart. These theorems are used in part II of this paper (unpublished), in which the authors prove the following: For a chart \(\Gamma\) with at most two crossings, if the closure of the surface braid represented by \(\Gamma\) is a disjoint union of spheres, then \(\Gamma\) is a ribbon chart.