Isotopy classes for 3-periodic net embeddings

Symmetric tangled Platonic polyhedra

Tangled tetrahedra, octahedra, cubes, icosahedra, and dodecahedra are generalizations of classical—untangled—Platonic polyhedra. Like the Platonic polyhedra, all vertices, edges, and faces are symmetrically equivalent. However, the edges of tangled polyhedra are curvilinear, or kinked, to allow entanglement, much like warps and wefts in woven fabrics. We construct the most symmetric entanglements of these polyhedra via assemblies of multistrand helices wound around edges of the conventional polyhedra; they are all necessarily chiral. The construction gives self-entangled chiral polyhedra and compound polyhedra containing catenated multiple tetrahedra or “generalized θ-polyhedra.” An unlimited variety of tangling is possible for any given topology. Related structures have been observed in synthetic materials and clathrin assemblies within cells.

Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols I_h, O_h, and T_d). Tangled Platonic {f, z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,… strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θ_z” polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

On Cayley graphs of {\bb Z}^4

The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components -1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

Isotopy classification of three-dimensional embedded nets

The article by Power et al. [Acta Cryst. (2020), A76 , 275–301 ] on the isotopy classification of crystal nets is discussed.