Remarks on the Vassiliev knot invariants coming from \(sl_ 2\)
The paper describes the combinatorial approach to Vassiliev's knot invariants via weight systems on chord diagrams for the Lie algebra \(sl_2\) with an invariant bilinear form \(Tr\) which can be interpreted as a trace. The main result (Theorem 6) gives a 3-term skein-type relation for so-called ``Japanese character diagrams'', JCD, which expresses \(W(X)\), \(X\) a JCD, by a sum of two such terms on JCD's with less vertices. The JCD's are introduced as a generalization of Bar-Natans ``Chinese character diagrams'' (CCD) [\textit{D. Bar-Natan}, On the Vassiliev knot invariants, Topology 34, 423-472 (1995)]. As a consequence of Theorem 6 several 6-term relations for chord diagrams are derived. The paper contains furthermore properties of the weight system of an arbitrary simple Lie algebra and the primitive part of the algebra of chord diagrams. The described approach depends on Kontsevich's construction of weight systems.
scientific article; zbMATH DE number 938860