Abstract

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 F be an A_k−1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let g be the quotient set of F under A_k-equivalence. We show that the set g forms an abelian group under a certain geometric operation. We define finite type invariants on 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 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 ≤ k − 1 takes the same value on them. The ‘if’ part does not hold for two-component links. Our result gives a suficient condition for spatial graphs to be C_k-equivalent.

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