Local moves on spatial graphs and finite type invariants

    Research output: Contribution to journalArticle

    9 Citations (Scopus)

    Abstract

    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
    Volume211
    Issue number1
    Publication statusPublished - 2003 Sep

    Fingerprint

    Spatial Graph
    Finite Type Invariants
    Equivalence
    Isotopy
    Finite Graph
    Vassiliev Invariants
    Equivalence relation
    Equivalence class
    Natural number
    Knot
    Abelian group
    Quotient
    If and only if
    Denote
    Imply
    Sufficient Conditions

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

    Local moves on spatial graphs and finite type invariants. / Taniyama, Kouki; Yasuhara, Akira.

    In: Pacific Journal of Mathematics, Vol. 211, No. 1, 09.2003, p. 183-200.

    Research output: Contribution to journalArticle

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