266 research outputs found

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

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    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'

    Finding k partially disjoint paths in a directed planar graph

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    The {\it partially disjoint paths problem} is: {\it given:} a directed graph, vertices r1,s1,…,rk,skr_1,s_1,\ldots,r_k,s_k, and a set FF of pairs {i,j}\{i,j\} from {1,…,k}\{1,\ldots,k\}, {\it find:} for each i=1,…,ki=1,\ldots,k a directed riβˆ’sir_i-s_i path PiP_i such that if {i,j}∈F\{i,j\}\in F then PiP_i and PjP_j are disjoint. We show that for fixed kk, 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,…,k}\{1,\ldots,k\} prescribing which of the riβˆ’sir_i-s_i paths are allowed to traverse this edge

    On traces of tensor representations of diagrams

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    Let TT be a set, of {\em types}, and let \iota,o:T\to\oZ_+. A {\em TT-diagram} is a locally ordered directed graph GG equipped with a function Ο„:V(G)β†’T\tau:V(G)\to T such that each vertex vv of GG has indegree ΞΉ(Ο„(v))\iota(\tau(v)) and outdegree o(Ο„(v))o(\tau(v)). (A directed graph is {\em locally ordered} if at each vertex vv, linear orders of the edges entering vv and of the edges leaving vv are specified.) Let VV be a finite-dimensional \oF-linear space, where \oF is an algebraically closed field of characteristic 0. A function RR on TT assigning to each t∈Tt\in T a tensor R(t)∈Vβˆ—βŠ—ΞΉ(t)βŠ—VβŠ—o(t)R(t)\in V^{*\otimes \iota(t)}\otimes V^{\otimes o(t)} is called a {\em tensor representation} of TT. The {\em trace} (or {\em partition function}) of RR is the \oF-valued function pRp_R on the collection of TT-diagrams obtained by `decorating' each vertex vv of a TT-diagram GG with the tensor R(Ο„(v))R(\tau(v)), and contracting tensors along each edge of GG, while respecting the order of the edges entering vv and leaving vv. In this way we obtain a {\em tensor network}. We characterize which functions on TT-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

    The Strong Arnold Property for 4-connected flat graphs

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    We show that if G=(V,E)G=(V,E) is a 4-connected flat graph, then any real symmetric VΓ—VV\times V matrix MM with exactly one negative eigenvalue and satisfying, for any two distinct vertices ii and jj, Mij<0M_{ij}<0 if ii and jj are adjacent, and Mij=0M_{ij}=0 if ii and jj are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric VΓ—VV\times V matrix XX with MX=0MX=0 and Xij=0X_{ij}=0 whenever ii and jj are equal or adjacent. (A graph GG is {\em flat} if it can be embedded injectively in 33-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

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    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
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