Relations about geometric characteristics of surface knots and invariants

Kamada定义图表来研究4空间中的嵌入曲面。图表是平面上的有向图，其每条边都用1到n-1的整数之一来标记，并且每个顶点都是满足某些条件的1度、4度或6度。1次顶点称为黑顶点，4次顶点称为交叉，6次顶点称为白顶点。图表之间存在c移动，使得c移动不会改变4空间中相关表面的环境同位素类别。如果一个图表可以通过c移动移动一个没有白色顶点的图表，那么它就是一个色带。Kamada证明了任何3图都是带状图。我们证明了如果任意n-图最多包含两个交叉点，并且它在4空间中的关联面是球的不相交并，则该图是带状图。为了证明这个定理，我们证明了任意最小广义n图包含恰好两个交叉点，那么该图包含至少4n-10个黑色顶点。众所周知，在4空间中任何图表的关联面都是一个球体，那么该图表恰好包含2n-2个黑色顶点。我们得到了想要的结果。为了研究黑色顶点的数量，我们定义了图表的缠结。我们发现了一些减少缠结白色顶点的条件。我们研究恰好包含4、5或7个白色顶点的图表。如果一个图表包含5或7个白色顶点，那么我们可以通过c移动来取消一些白色顶点。

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.