The combined first-order and second-order motions graph for genera 9 -14 are shown in Fig. with the graph edges coloured as first order: blue; and second order: red. Note, that the topology of this combined graph is that of a 2-torus. Careful inspection of this graph shows that it is composed of six sets of rings ("cycles") that are each composed of six species. The firstorder motions and second-order motions graphs are shown in Fig. and Fig. , respectively. The latter two graphs have the topology of a 2-torus as, by the modular structure theorem, each are subgraphs of the combined graph in Fig. . This second-order motions graph (Fig. ) forms a disconnected graph which indicates that second-order motions cannot interconvert species between the separate subgraph components Fig. Toroidal desmotropic reaction graphs of genera 9 -14 for the 𝒮 3 ℬ 2 class. All graph vertices are labeled by the bonder connectivity (omitting the κ i symbols) and edges are coloured by motions type (blue: first order; red: second order). a) Combined first-and second-order motions graph laid out to show the toroidal form. First-order and second-order motions indicated by blue and red graph edges, respectively. b) First-order motions graph. c) Second-order motions graph. This is a disconnected graph. Fig. Interpolated approximation of the DFT-calculated potential-energy surface for tautomerism in free-base subporphyrin partitioned as 𝓢 𝟑 𝓑 𝟐 using genera 9 -14. Surface generated from calculated energies for representative structures for each genus. a) 2D-periodic rendering of the 2-toroidal PES. All species located on the surface are labeled by their species number (see Supplementary Information for details). The piecewise reaction coordinates of each bonder are periodic with replicate configuration genera shown in gray (top row and right edge). Desmotropic reactions are indicated as first order (solid black lines) and two second order networks (red dashed and red dotted lines). b) 3D rendering of the PES. c) The six genera with representative structures and their compact symbols indicating PES character. Stereochemistries are only demonstrative.