Local moves on spatial graphs and finite type invariants

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    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 script F Sign be an Ak-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 Ak-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 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.

    Original languageEnglish
    Pages (from-to)183-200
    Number of pages18
    JournalPacific Journal of Mathematics
    Issue number1
    Publication statusPublished - 2003 Sep


    ASJC Scopus subject areas

    • Mathematics(all)

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