On traces of tensor representations of diagrams

Let $T$ be a set, of {\em types}, and let $\iota,o:T\to\oZ_+$. A {\em $T$-diagram} is a locally ordered directed graph $G$ equipped with a function $\tau:V(G)\to T$ such that each vertex $v$ of $G$ has indegree $\iota(\tau(v))$ and outdegree $o(\tau(v))$. (A directed graph is {\em 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 $\oF$-linear space, where $\oF$ is an algebraically closed field of characteristic 0. A function $R$ on $T$ assigning to each $t\in T$ a tensor $R(t)\in V^{*\otimes \iota(t)}\otimes V^{\otimes o(t)}$ is called a {\em tensor representation} of $T$. The {\em trace} (or {\em partition function}) of $R$ is the $\oF$-valued function $p_R$ 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 {\em 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.