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

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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