Local moves on spatial graphs and finite type invariants

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 sufficient condition for spatial graphs to be C k -equivalent.