In the previous chapter we saw that the study of Vassiliev knot invariants, at least complex-valued, is largely reduced to the study of the algebra of chord diagrams. Here we introduce two different types of diagrams representing elements of this algebra, namely closed Jacobi diagrams and open Jacobi diagrams. These diagrams provide better understanding of the primitive space PA and bridge the way to the applications of the Lie algebras in the theory of Vassiliev invariants; see Chapter 6.

The name Jacobi diagrams is justified by a close resemblance of the basic relations imposed on Jacobi diagrams (STU and IHX) to the Jacobi identity for Lie algebras.

Closed Jacobi diagrams

Definition 5.1.

A closed Jacobi diagram (or, simply, a closed diagram) is a connected trivalent graph with a distinguished simple oriented cycle, called Wilson loop, and a fixed cyclic order of half-edges at each vertex not on the Wilson loop. Half the number of the vertices of a closed diagram is called the degree, or order, of the diagram. This number is always an integer.

Remark 5.2. Some authors (see, for instance, Habegger and Masbaum 2000) also include the cyclic order of half-edges at the vertices on the Wilson loop into the structure of a closed Jacobi diagram; this leads to the same theory.

Remark 5.3. A Jacobi diagram is allowed to have multiple edges and hanging loops, that is, edges with both ends at the same vertex.