On the Vassiliev knot invariants.

*(English)*Zbl 0898.57001This 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) |