Bar-Natan, Dror On the Vassiliev knot invariants. (English) Zbl 0898.57001 Topology 34, No. 2, 423-472 (1995). 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. Reviewer: Alexander Frey (Berlin) Cited in 16 ReviewsCited in 372 Documents 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 PDFBibTeX XMLCite \textit{D. Bar-Natan}, Topology 34, No. 2, 423--472 (1995; Zbl 0898.57001) 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.