{
  "id": "math/0106173",
  "version": "v1",
  "published": "2001-06-20T18:08:56.000Z",
  "updated": "2001-06-20T18:08:56.000Z",
  "title": "Local moves on spatial graphs and finite type invariants",
  "authors": [
    "Kouki Taniyama",
    "Akira Yasuhara"
  ],
  "comment": "LaTeX, 18 pages with figures, to appear in Pacific Journal of Mathematics",
  "categories": [
    "math.GT"
  ],
  "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 ${\\cal F}$ be an $A_{k-1}$-equivalence class of the embeddings of a finite graph into the 3-sphere. Let ${\\cal G}$ be the quotient set of ${\\cal F}$ under $A_k$-equivalence. We show that the set ${\\cal G}$ forms an abelian group under a certain geometric operation. We define finite type invariants on ${\\cal F}$ 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 ${\\cal F}$, 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 $\\leq k-1$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-06-20T18:08:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M15"
    ],
    "keywords": [
      "spatial graphs",
      "local moves",
      "equivalence",
      "finite graph",
      "define finite type invariants"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......6173T"
    }
  }
}