The algebra of 3-graphs Γ, introduced in Duzhin et al. (1998), is related to the diagram algebras C and B. The difference between 3-graphs and closed diagrams is that 3-graphs do not have a distinguished cycle (Wilson loop); neither do they have univalent vertices, which distinguishes them from open diagrams. Strictly speaking, there are two different algebra structures on the space of 3-graphs, given by the edge (Section 7.2) and the vertex (Section 7.3) products. The space Γ is closely related to the Vassiliev invariants in several ways:

1. • The vector space Γ is isomorphic to the subspace P2 of the primitive space P ⊂ C spanned by the connected diagrams with two legs (Section 7.4.1).

2. • The algebra Γ acts on the primitive space P in two ways, via the edge, and via the vertex products (see Sections 7.4.1 and 7.4.2). These actions behave nicely with respect to Lie algebra weight systems (see Chapter 6); as a consequence, the algebra Γ is as good a tool for the proof of existence of non-Lie-algebraic weight systems as the algebra Γ in Vogel's original approach (Section 7.6.4).

3. • The vector space Γ describes the combinatorics of finite type invariants of integral homology 3-spheres in much the same way as the space of chord diagrams describes the combinatorics of Vassiliev knot invariants. This topic, however, lies outside of the scope of our book and we refer an interested reader to Ohtsuki (2002)

4. Unlike L and B, the algebra Γ does not have any natural coproduct.