Study of local moves and invariants for knots and virtual knots

我们打一个结，并确定它的端点。没有端点的图形在拓扑学中称为“结”。R^3中的图称为空间图。如果我们在R^3中连续地将两个纽结或空间图变换成另一个纽结或空间图，并且使相同的数学值对应于相同的纽结或空间图，则认为这两个纽结或空间图是相同的。这个值称为不变量。有限型不变量是一个不变量，它将所有的纽结不变量按顺序分类。在本研究中，我们定义了空间图的一个新的不变量，并研究了它的性质。对于结点，我们引入了度量或复形的概念，并研究了结点空间中有限型不变量的性质。

We tie a knot and identify the end points of it. That figure having no end points is called "a knot" in Topology. A graph in R^3 is called a spatial graph. Two knots or spatial graphs are thought to be same if we transform one into another continuously in R^3 and the same mathematical value is made to correspond to the same knots or spatial graphs. The value is called an invariant. The finite type invariant is an invariant that classifies all the knot invariants by order. In this study, we define a new invariant for spatial graphs and investigate the property of it. For knots, we introduce the notion of a metric or a complex and we investigate the property of finite type invariants in the knot space.