On traces of tensor representations of diagrams

Let T be an (abstract) set of types, and let (unknown symbol), o : T -> Z(+). A T-diagram is a locally ordered directed graph G equipped with a function tau : V (G) -> T such that each vertex v of G has indegree (unknown symbol)(tau(v)) and outdegree o(tau(v)). (A directed graph is locally ordered if at each vertex v, linear orders of the edges entering v and of the edges leaving v are specified.)

Let V be a finite-dimensional F-linear space, where F is an algebraically closed field of characteristic 0. A function R on T assigning to each t is an element of T a tensor R(t) is an element of V*(circle times l(t)) circle times V-circle times o(t) is called a tensor representation of T. The trace (or partition function) of R is the F-valued function pR on the collection of T-diagrams obtained by 'decorating' each vertex v of a T-diagram G with the tensor R(tau(v)), and contracting tensors along each edge of G, while respecting the order of the edges entering v and leaving v. In this way we obtain a tensor network.

We characterize which functions on T-diagrams are traces, and show that each trace comes from a unique 'strongly nondegenerate' tensor representation. The theorem applies to virtual knot diagrams, chord diagrams, and group representations.