TY  - JOUR
A1  - Hyde, Stephen T.
A1  - Evans, Myfanwy E.
T1  - Symmetric tangled Platonic polyhedra
JF  - Proceedings of the National Academy of Sciences of the United States of America
N2  - 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 (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 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 Schonflies 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 "theta(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.
KW  - regular polyhedra
KW  - compound polyhedra
KW  - helicates
KW  - metal-organic
KW  - frameworks
KW  - clathrin
Y1  - 2022
U6  - https://doi.org/10.1073/pnas.2110345118
SN  - 0027-8424
SN  - 1091-6490
VL  - 119
IS  - 1
PB  - National Acad. of Sciences
CY  - Washington
ER  - 
TY  - JOUR
A1  - Evans, Myfanwy E.
A1  - Hyde, Stephen T.
T1  - Symmetric Tangling of Honeycomb Networks
JF  - Symmetry
N2  - Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.
KW  - tangles
KW  - knots
KW  - networks
KW  - periodic entanglement
KW  - molecular weaving
KW  - graphs
Y1  - 2022
U6  - https://doi.org/10.3390/sym14091805
SN  - 2073-8994
VL  - 14
SP  - 1
EP  - 13
PB  - MDPI
CY  - Basel, Schweiz
ET  - 9
ER  -