Chapter 3 Colorings

A colored graph is a pair \((G,c)\) of a graph \(G\) and a coloring function \(c : V(G) \to \{\pm1\}\).

3.1 Groupoid of colored graphs

Define isomorphisms between colored graphs \((G,c)\) and \((G',c')\) as pairs \((f,g)\) of a graph isomorphism \(f: G \to G'\) and a function \(g : E(G') \to \{\pm1\}\) such that for any vertex \(v\in V(G)\) with edges of \(f(v)\) denoted \(i',j',k'\) the relation \(c'(fv) / c(v) = g(i') g(j') g(k')\) holds. In particular if \(f\) is the identity and we regard the colorings \(c,c'\) as \(0\)-chains, then the respective morphisms is the set of \(1\)-chains whose boundary is the ratio of \(c'\) and \(c\), so the set of automorphisms is principal homogeneous over the first relative homology group \(H_1(G,∂G;\{\pm1\})\). The composition of \((f,g) : (G,c) \to (G',c')\) and \((f',g') : (G',c') \to (G'',c'')\) is \((f'\circ f, g'')\) where \(g''(i'') := g'(i'') \cdot \prod_{f'(i') = i''} g'(i')\).

Generators for the groupoid of colored graphs can be chosen as morphisms \((f,g) : (G,c) \to (G',c')\) with either \(c = c' \circ f, g = 1\) or \(f\) being identity and \(g\) a coloring of a single edge.

3.2 Functor of quantum Clebsch-Gordan polytopes

Define \(P(G,c)\) as the convex envelope in \(S(G)\) of rays \(p(v,s)\) with \(s(i) s(j) s(k) = c(v)\). Note that \(P(G) = P(G,c_0)\), where \(c_0\) is the constant coloring \(c_0(v) = +1 \forall v\).

Any isomorphism \((f,g): (G,c) \to (G',c')\) of colored graphs naturally induces an isomorphism \(P(f,g) : P(g,c) \to P(G',c')\) of the associated convex polytopes. For example, the generators of the first kind correspond to “permutations” and the generators of the second kind correspond to linear transformations that send \(e(i)\) to \(-e(i)\) for the chosen edge \(i\) and preserve all other basic vectors.

3.3 Colored combinatorial non-abelian Torelli theorem

Theorem 3.1 Association \(P\) is a full functor from the groupoid of colored graphs of principal series to the groupoid of convex polytopes and affine isomorphisms. That is, any affinely linear isomorphism between polytopes \(P(G,c)\) and \(P(G',c')\) is induced by a uniquely defined isomorphism of colored graphs.

Proof. The steps of the algorithm above can be applied without modifications.

Then for any of \(2^{\dim S}\) bases of \(S\) with given coordinate axes \(±e(i)\) the coloring of the vertices of the graph is well- and uniquely defined by the parity of the number of minus signs in the coordinates of the respective rays in the given basis. A unique choice of signs corresponds to the original coloring, all other choices produce homologically equivalent colorings, unique invariants of colorings are total parity of vertices in every leafless connected component. It suffices to notice that the change of the sign of a single vector in the base corresponds to simultaneous change of the color of the two vertices adjacent to this edge.