Local moves on spatial graphs and finite type invariants

الأصل · EN

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.