### Abstract

Let Ẑ_{f} be the universal Vassiliev-Kontsevich invariant for framed links in [13], which is a generalization of Kontsevich's invariant in [10, 1]. Let K be a framed knot and K^{(r)} be its r-parallel. Then we show Ẑ_{f}(K^{(r)}) = Δ_{(r)}(Ẑ_{f}(K)), where Δ_{(r)} is an operation of chord diagrams which replace the Wilson loop by r copies. We calculate the values of Ẑ_{f} of the Hopf links and the change of Ẑ_{f} under the Kirby moves. An explicit formula of an important normalization factor, which is the value of the trivial knot, in the universal enveloping algebra U(g) of any Lie algebra is given.

Original language | English |
---|---|

Pages (from-to) | 271-291 |

Number of pages | 21 |

Journal | Journal of Pure and Applied Algebra |

Volume | 121 |

Issue number | 3 |

Publication status | Published - 1997 Oct 10 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Pure and Applied Algebra*,

*121*(3), 271-291.

**Parallel version of the universal Vassiliev-Kontsevich invariant.** / Le, Thang T Q; Murakami, Jun.

Research output: Contribution to journal › Article

*Journal of Pure and Applied Algebra*, vol. 121, no. 3, pp. 271-291.

}

TY - JOUR

T1 - Parallel version of the universal Vassiliev-Kontsevich invariant

AU - Le, Thang T Q

AU - Murakami, Jun

PY - 1997/10/10

Y1 - 1997/10/10

N2 - Let Ẑf be the universal Vassiliev-Kontsevich invariant for framed links in [13], which is a generalization of Kontsevich's invariant in [10, 1]. Let K be a framed knot and K(r) be its r-parallel. Then we show Ẑf(K(r)) = Δ(r)(Ẑf(K)), where Δ(r) is an operation of chord diagrams which replace the Wilson loop by r copies. We calculate the values of Ẑf of the Hopf links and the change of Ẑf under the Kirby moves. An explicit formula of an important normalization factor, which is the value of the trivial knot, in the universal enveloping algebra U(g) of any Lie algebra is given.

AB - Let Ẑf be the universal Vassiliev-Kontsevich invariant for framed links in [13], which is a generalization of Kontsevich's invariant in [10, 1]. Let K be a framed knot and K(r) be its r-parallel. Then we show Ẑf(K(r)) = Δ(r)(Ẑf(K)), where Δ(r) is an operation of chord diagrams which replace the Wilson loop by r copies. We calculate the values of Ẑf of the Hopf links and the change of Ẑf under the Kirby moves. An explicit formula of an important normalization factor, which is the value of the trivial knot, in the universal enveloping algebra U(g) of any Lie algebra is given.

UR - http://www.scopus.com/inward/record.url?scp=0031563734&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031563734&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031563734

VL - 121

SP - 271

EP - 291

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 3

ER -