Study of local moves and invariants for knots and virtual knots

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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.