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**Knot polynomials and Vassiliev’s invariants.**
*(English)*
Zbl 0812.57011

The main result of the paper offers an interpretation of the Jones’ polynomial (and further polynomials created in its wake) in terms of more classical concepts of topology. V. A. Vassiliev introduced in ‘Theory of singularities and its applications’ [Adv. Sov. Math. 1, 23-69 (1990; Zbl 0727.57008)] (a virtually identical version is Chapter V of [V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications (1992; Zbl 0762.55001)]) new invariants of knots based on a spectral sequence associated to the space of smooth maps from the circle to the 3-sphere. The authors prove that after a change of variables which transforms the knot polynomials into power series the coefficients of these series are Vassiliev type invariants. In order to prove this they give an axiomatic description of Vassiliev invariants and then show that the coefficients of the knot power series satisfy the axioms. This axiomatic approach has proved extremely useful, making Vassiliev invariants much more accessible.

Reviewer: E.Vogt (Berlin)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

55T99 | Spectral sequences in algebraic topology |

58D10 | Spaces of embeddings and immersions |

57R40 | Embeddings in differential topology |

57R45 | Singularities of differentiable mappings in differential topology |

### Keywords:

Vassiliev invariants; Jones’ polynomial; space of smooth maps from the circle to the 3-sphere; knot polynomials
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\textit{J. S. Birman} and \textit{X.-S. Lin}, Invent. Math. 111, No. 2, 225--270 (1993; Zbl 0812.57011)

### References:

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