Characterizing partition functions of the edge-coloring model by rank growth

We characterize which graph invariants are partition functions of an edge-coloring model over the complex numbers, in terms of the rank growth of associated `connection matrices'

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

Finding k partially disjoint paths in a directed planar graph

The {\it partially disjoint paths problem} is: {\it given:} a directed graph, vertices $r_1,s_1,\ldots,r_k,s_k$, and a set $F$ of pairs $\{i,j\}$ from $\{1,\ldots,k\}$, {\it find:} for each $i=1,\ldots,k$ a directed $r_i-s_i$ path $P_i$ such that if $\{i,j\}\in F$ then $P_i$ and $P_j$ are disjoint. We show that for fixed $k$, this problem is solvable in polynomial time if the directed graph is planar. More generally, the problem is solvable in polynomial time for directed graphs embedded on a fixed compact surface. Moreover, one may specify for each edge a subset of $\{1,\ldots,k\}$ prescribing which of the $r_i-s_i$ paths are allowed to traverse this edge

The Strong Arnold Property for 4-connected flat graphs

We show that if $G=(V,E)$ is a 4-connected flat graph, then any real symmetric $V\times V$ matrix $M$ with exactly one negative eigenvalue and satisfying, for any two distinct vertices $i$ and $j$, $M_{ij}<0$ if $i$ and $j$ are adjacent, and $M_{ij}=0$ if $i$ and $j$ are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric $V\times V$ matrix $X$ with $MX=0$ and $X_{ij}=0$ whenever $i$ and $j$ are equal or adjacent. (A graph $G$ is {\em flat} if it can be embedded injectively in $3$-dimensional Euclidean space such that the image of any circuit is the boundary of some disk disjoint from the image of the remainder of the graph.) This applies to the Colin de Verdi\`ere graph parameter, and extends similar results for 2-connected outerplanar graphs and 3-connected planar graphs

On the existence of real R-matrices for virtual link invariants

We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon-Wales theorem on the decomposition of Brauer algebras, and the Procesi-Schwarz theorem on inequalities for closed orbits