Relations about geometric characteristics of surface knots and invariants

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Kamada define charts to investigate embedded surfaces in 4-space. A chart is a oriented graph in the plane such that each edge are labeled by one of integers from 1 to n-1, and each vertex is degree 1, 4 or 6 satisfying some conditions. A vertex of degree 1 is called a black vertex, a vertex of degree 4 is a crossing, and a vertex of degree 6 is a white vertex. There are C-moves between charts such that C-moves do not change the ambient isotopy classes of the associated surfaces in 4-space. A chart is a ribbon if it can be moved a chart without white vertices by C-moves.Kamada showed that any 3-chart is a ribbon chart. We show that if any n-chart contains at most two crossings, and if its associated surface in 4-space is a disjoint union of spheres, then the chart is a ribbon chart. To show this theorem, we show that any minimal generalized n-chart contains exactly two crossings, then the chart contains at least 4n-10 black vertices. It is well known that the associated surface of any chart in 4-space is one sphere, then the chart contains exactly 2n-2 black vertices. We have the desired result. To investigate the number of black vertices, we define tangles for charts. We found some condition reducing white vertices for tangles.We investigate charts containing exactly 4, 5 or 7 white vertices. If a chart contains exactly 5 or 7 white vertices, then we can cancel some white vertex by C-moves.