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Relations about geometric characteristics of surface knots and invariants

表面结几何特征与不变量的关系

基本信息

批准号：
16540082
负责人：
SHIMA Akiko
金额：
$ 1.99万
依托单位：
Tokai University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540082/
关键词：
topology surface link chart surface braid surface knot crossing minimal bigon triple point

项目摘要

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.

Kamada定义了图表来研究4维空间中的嵌入表面。图是平面上的有向图，其中每条边由1到n-1中的一个整数标记，每个顶点满足一定条件的度为1、4或6。次数为1的顶点称为黑色顶点，次数为4的顶点为交叉点，次数为6的顶点为白色顶点。在图表之间存在C-移动，使得C-移动不会改变4-空间中相关曲面的环境同位素类。如果图表可以通过C-move移动到没有白色顶点的图表，那么它就是带状的。Kamada证明了任何3-图表都是带状的。证明了：如果任一n-图至多包含两个交叉点，且其在4-空间中的相关曲面是球面的不相交并，则该图是带状图。为了证明这一定理，我们证明了任何极小广义n-图恰好包含两个交叉点，则该图至少包含4N-10个黑色顶点。众所周知，4维空间中任一图的关联曲面都是一个球面，则该图恰好包含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。我们发现了一些关于正切的白色顶点减少的条件，我们研究了恰好包含4，5或7个白色顶点的图。如果一个图恰好包含5个或7个白色顶点，那么我们可以通过C-Moves来消除一些白色顶点。