Combinatorial non-abelian Torelli theorem
Abstract
We prove that a (colored) trivalent graph can be recovered from (the polar dual of) the associated quantum Clebsch-Gordan polytope and that any isomorphism between such polytopes is induced by a unique properly defined isomorphism of underlying colored graphs.
Chapter 1 Graphs and polytopes
To a graph \(G\) with the set \(V(G)\) of trivalent vertices associate a vector space \(S(G)\) with a basis \(e(i)\) enumerated by (non-oriented) edges of \(G\), and a polytope \(P(G)\) there given as a convex envelope of the vectors \[\begin{equation} p(v,s) = s(i) e(i) + s(j) e(j) + s(k) e(k) = ±e(i)±e(j)±e(k), \end{equation}\] where the edges \(i,j,k\) are outgoing from a single vertex \(v\) and \(s(i),s(j),s(k)\in\{\pm1\}\) with \(s(i)s(j)s(k)=1\).
Let us prove that the polytopes \(P(G)\) to \(P(G')\) can be mapped to each other by a pair of (real) affine transformations if and only if the graphs \(G\) and \(G'\) are isomorphic.
1.1 Graphs of principal series
The graph \(G\) is said to be of principal series if it satisfies the following:
Every connected component of \(G\) has at least \(5\) trivalent vertices.
Graph \(G\) has no loops.
Graph \(G\) has no double edges.
Graph \(G\) has no leaves.