Local moves on spatial graphs and finite type invariants

We define $A_k$-moves for embeddings of a finite graph into the 3-sphere for each natural number $k$. Let $A_k$-equivalence denote an equivalence relation generated by $A_k$-moves and ambient isotopy. $A_k$-equivalence implies $A_{k-1}$-equivalence. Let ${\cal F}$ be an $A_{k-1}$-equivalence class of the embeddings of a finite graph into the 3-sphere. Let ${\cal G}$ be the quotient set of ${\cal F}$ under $A_k$-equivalence. We show that the set ${\cal G}$ forms an abelian group under a certain geometric operation. We define finite type invariants on ${\cal F}$ of order $(n;k)$. And we show that if any finite type invariant of order $(1;k)$ takes the same value on two elements of ${\cal F}$, then they are $A_k$-equivalent. $A_k$-move is a generalization of $C_k$-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to $C_k$-move and ambient isotopy if and only if any Vassiliev invariant of order $\leq k-1$ takes the same value on them. The ` if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be $C_k$-equivalent.Comment: LaTeX, 18 pages with figures, to appear in Pacific Journal of Mathematic