Quasilocal defects in regular planar networks: Categorization for molecular cones

Graphical networks are cast into structural equivalence classes, with special focus on ones related to two‐dimensional regular translationally symmetric nets (or lattices). A quasilocal defect in a regular net is defined as consisting of a finite subnet surrounded outside this region by an infinitely extended network of which arbitrary, simply connected regions are isomorphic to those of the regular undefected net. The global equivalence classes for such quasilocal defects are identified by a “circum‐matching” characteristic. One or more such classes are identified as a “turn” number, or equivalently as a discrete “combinatorial curvature” κ, which associates closely to the geometric Gaussian curvature of “physically reasonable” embeddings of the net in Euclidean space. Then for positive κ, geometric cones result; for κ = 0, the network is flat overall; and for negative κ, fluted or crenalated cones result. As κ orqvaries through its discrete range, the number of defect classes varies between 1 and ∞ and repeats with a period depending on the parent regular net. For the square‐planar net, the numbers of defect classes at succeeding turn numbers (q) starting atq= 0 are ∞, 2, 3, 2, repeating with a period of 4. For the hexagonal and triangular nets, the numbers of classes at suceedingqstarting atq= 0 are ∞, 1, 2, 2, 2, 1, repeating with a period of 6. A further refinement of the classes of quasilocal defects breaks these classes up into “irrotational” subclasses, as are relevant for multiwall cones. The subclasses are identified via a “quasispin” characteristic, which is conveniently manipulatable for the categorization of multiwall cones. Besides the development of the overall comprehensive topo‐combinatoric categorization scheme for quasilocal defects, some consequences are briefly indicated, and combining rules for the characteristics of pairs of such defects are briefly considered.