On traces of tensor representations of diagrams

Let T be an (abstract) set of types  , and let ι,o:T→Z+ι,o:T→Z+. A T-diagram is a locally ordered directed graph G   equipped with a function τ:V(G)→Tτ:V(G)→T such that each vertex v of G   has indegree ι(τ(v))ι(τ(v)) and outdegree o(τ(v))o(τ(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 FF-linear space, where FF is an algebraically closed field of characteristic 0. A function R on T   assigning to each t∈Tt∈T a tensor R(t)∈V⁎⊗ι(t)⊗V⊗o(t)R(t)∈V⁎⊗ι(t)⊗V⊗o(t) is called a tensor representation of T. The trace (or partition function) of R   is the FF-valued function pRpR on the collection of T-diagrams obtained by ‘decorating’ each vertex v of a T-diagram G   with the tensor R(τ(v))R(τ(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.