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**On the Vassiliev knot invariants.**
*(English)*
Zbl 0898.57001

This article which appeared in 1995 can by today be considered one of the “classical” articles in the topic of Vassiliev invariants; it is referred to by almost any author treating the subject. This is due to the fact that the article presents a large collection of material – most of which is due to the author, to Kontsevich and to Birman and Lin – as well as a lot of important basic notions such as weight system, 4-T-relation, STU relation and others. Among the questions treated are:

– The definition of Vassiliev invariants and of weight systems. It is shown that any Vassiliev invariant leads to a weight system and the theorem of Kontsevich is presented which states that over the real numbers any weight system can be integrated up as to yield a Vassiliev invariant, thus over \(\mathbb R\), the notions of weight system and Vassiliev invariant are essentially the same.

– The algebra of chord diagrams, its Hopf-algebra structure, different descriptions of this algebra, one of which is especially suited for dealing with primitive elements.

– The connection between the classical knot polynomials and Vassiliev invariants.

– How to construct weight systems from Lie algebras.

A lot of exercises and conjectures are given. This article still gives a good introduction to the subject of Vassiliev invariants.

– The definition of Vassiliev invariants and of weight systems. It is shown that any Vassiliev invariant leads to a weight system and the theorem of Kontsevich is presented which states that over the real numbers any weight system can be integrated up as to yield a Vassiliev invariant, thus over \(\mathbb R\), the notions of weight system and Vassiliev invariant are essentially the same.

– The algebra of chord diagrams, its Hopf-algebra structure, different descriptions of this algebra, one of which is especially suited for dealing with primitive elements.

– The connection between the classical knot polynomials and Vassiliev invariants.

– How to construct weight systems from Lie algebras.

A lot of exercises and conjectures are given. This article still gives a good introduction to the subject of Vassiliev invariants.

Reviewer: Alexander Frey (Berlin)

### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

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

### Keywords:

Chinese character; marked surface; Lie algebra; Kontsevich integral; primitive element; weight system; 4-T-relation; STU relation; Vassiliev invariant; algebra of chord diagrams### Software:

OEIS
Full Text:
DOI

### Online Encyclopedia of Integer Sequences:

Dimension of space of weight systems of chord diagrams.Dimension of space of Vassiliev knot invariants of order n.

Number of circular chord diagrams with n chords, up to rotational symmetry.

Dimension of primitive Vassiliev knot invariants of order n.

Number of chord diagrams with n chords; number of pairings on a necklace.

Conjectured dimensions of spaces of weight systems of chord diagrams.

Conjectured numbers of Vassiliev invariants of knots.

Partial sums of A001935; at one time this was conjectured to agree with A007478.

Number of chord diagrams of degree n with an isolated chord.

Number of chord diagrams of degree n with an isolated chord of length 1.