State-Sum Iwarlants of Surface-knot in 4-space

设F是四维空间中的嵌入曲面。我们称之为曲面纽结。我们考虑F按p（x，y，z，w）=（x，y，z）的投影。那么，p（F）有一个片的交。三个片的相交称为三重点我们知道许多不可定向的表面结确定最小三重点数。然而，我们还没有确定的最小三重点号码定向曲面结佐藤表明，没有表面的三重点号码为1。Kamada证明了对任意数n，存在曲面纽结使得F的三重点数大于n。我们确定了二捻纺三叶纱的三相点数为4，三捻纺三叶纱的三相点数为6。第一个结果是通过使用新的不变量，称为状态和不变量。为了证明第一个结果，我们证明了如果F是一个曲面纽结，使得F的三重点数为3或小于3，则F的状态和不变量的值为a ...更多信息 n整数。另一方面，2捻纺三叶形的状态和不变量的值不是整数。而且，2捻纺三叶有四个三重点的投影。因此，2捻纺三叶草的最小三相点数为4。第二个结果是用另一个量子点S_4得到的态太阳不变量给出的。我们做了一个至少需要三个点的方程。从而得到三捻三叶纱的最小三相点数为6。表面结的例子比经典结的例子少。研究了满足一定条件的面向平面的标号图。该图表示为4-球和4-空间中的嵌入曲面。Kamada表明，3-chart是C-move等价于没有顶点的6度图。6次顶点对应于三重点C-移动意味着局部移动，它不会改变从图表中获得的嵌入表面的环境同位素类别。Nagase和Hirota证明了最多有一个交叉点的4-图是C-移动等价于没有6度顶点的图。交叉点是4度顶点。我们证明了，如果一个4-图最多有两个交叉和最多六个顶点的度1，那么图是C-move等价于一个图没有顶点的度6。如果4-图表示为4-空间中的一个嵌入球体，则该图有六个度为6的顶点。少

Let F be an embedded surface in 4-space. We call it surface knot We consider the projection of F by p(x,y,z,w)=(x,y,z). Then, p(F) has an intersection of sheets. The intersection of three sheets is called a triple point We know many non-orientable surface knots detemine the minimal triple point numbers. However, we had not determined the minimal triple point numbers of oriented surface knot Satoh showed that there is not surface knots of the triple point number one. Kamada showed that for any number n, there exists a surface knot such that the triple point number of F is grater than n. We determine that the triple point number of the 2-twist-spun trefoil is four and that the triple point number of the 3-twist-spun trefoil is six. The first result is shown by using new invariant, called state-sum invariants. To show the first result, we show that if F is a surface knot such that the triple point number of F is three or less that three, then the value of the state sum invariant of F is a … More n integer. On the other hand, the value of the state sum invariant of the 2-twist-spun trefoil is not an integer. And, the 2-twist-spun trefoil has the projection with four triple points. So, the 2-twist spun trefoil has the minimal triple point number four. The second result is shown by using the state-sun invariant obtained from anther quandle, S_4.I made a program for calculation. We made an equation which needs at least triple points. Then we obtain the 3-twist spun trefoil has the minimal triple point number six. We have less examples of surface knots than examples of classical knots. We research charts which is the planer oriented labeled graph satisfying some conditions. This graph is represented an embedded surface in 4-ball and 4-space. Kamada showed that 3-chart is C-move equivalent to a chart without vertices of degree 6. The vertex of degree 6 is corresponded to a triple point C-move means the local move which does not change ambient isotopy classes of the embedded surface obtained from charts. Nagase and Hirota show that 4-chart with at most one crossing is C-move equivalent to a chart without vertices of degree 6. A crossing is a vertex of degree 4. We show that if a 4-chart is with at most two crossing and at most six vertices of degree one, then the chart is C-move equivalent to a chart without vertices of degree 6. If the 4-chart is represent to one embedded sphere in 4-space, then the chart has six vertices of degree 6. Less