A Jacobi diagram in knot theory is a finite graph, possibly disconnected and allowing multiple edges and loops, whose vertices are incident with either one or three edge ends, together with a cyclic order of the three edge ends at every trivalent vertex. A loop contributes two edge ends at its vertex. In the convention used for the algebra of Chinese characters, every connected component has at least one univalent vertex, called a leg. These diagrams have no distinguished supporting circle or line (Bar-Natan 1995, author version, p. 8; Chmutov and Duzhin 2005, p. 3).
The degree of a Jacobi diagram is half its total number of vertices. This is an integer, since the sum of its vertex degrees is even. The empty diagram has degree zero. The diagram consisting of a single edge and its two legs has degree one.
Attaching legs to an oriented supporting circle or line gives diagrams on a skeleton. The cyclic orders at trivalent vertices are part of the data and need not be induced by a planar drawing. Bare Jacobi diagrams modulo the relations defining the algebra of Chinese characters are distinct from diagrams attached to a skeleton.
