Topological coordinates for toroidal structures

Physically realistic 3D geometries for toroidal trivalent networks can be produced from graph theoretical information alone, using the eigenvectors resulting from diagonalisation of the vertex adjacency matrix. Arguments from the problem of a quantum particle constrained to move on a surface show that three vectors suffice for zero-genus spherical cages, whereas four are needed for decorations of surfaces with genus 1 (in contrast to previous suggestions). Solutions for the problems arising from the systematic high degeneracies in the spectra of polyhex tori are proposed.