Properties of minimal charts and their applications IX: charts of type [formula omitted].

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let Γ be a chart, and we denote by Γ m the union of all the edges of label m. A chart Γ is of type (4 , 3) if there exists a label m such that w (Γ) = 7 , w (Γ m ∩ Γ m + 1) = 4 , w (Γ m + 1 ∩ Γ m + 2) = 3 where w (G) is the number of white vertices in G. In this paper, we prove that there is no minimal chart of type (4 , 3). [ABSTRACT FROM AUTHOR]