Local Moves on Spatial Graphs and Finite Type Invariants
نویسندگان
چکیده
We define Ak-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let Ak-equivalence denote an equivalence relation generated by Ak-moves and ambient isotopy. Ak-equivalence implies Ak−1-equivalence. Let F be an Ak−1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let G be the quotient set of F under Ak-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 Ak-equivalent. Ak-move is a generalization of Ck-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to Ck-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 Ck-equivalent.
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