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 script F Sign be an A_k-1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let script G Sign be the quotient set of script F Sign under A_k-equivalence. We show that the set script G Sign forms an abelian group under a certain geometric operation. We define finite type invariants on script F Sign 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 script F Sign, 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 ≤ 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.