All toroidal embeddings of polyhedral graphs in 3-space are chiralw

The interplay between structural chemistry, graph theory and knot theory is a rich area for chemists and mathematicians alike. The founder of topology, Johann Listing, noted the possibility of chiral knots in 1847; a year later Pasteur published his landmark paper on the optical activity of ammonium tartrate crystals. Indeed, knot theory began following early studies on molecular isomerism. More recently, the investigation of chirality and knotting in organic molecules and DNA complexes using graph theoretical techniques has led to a number of fruitful results. In particular, chemists now speak of topological chirality, characterised by non-superimposable 3D mirror images of graphs among all possible embeddings of the graph that preserve ambient isotopy. Entangled (and, more specifically knotted) graphs or nets have generated considerable interest in synthetic organic chemistry. More recently, the possibility of entanglement and knotting has been raised in extended framework coordination polymer materials. This paper describes a strong interplay between tangled nets and the notion of chirality. We look at simple graphs whose topologies are those of the edges of polyhedra, such as the network of edges in an octahedron or icosahedron. (We define these ‘polyhedral graphs’ more precisely below.) The simplest realisations of polyhedral graph topologies in space form a network of edges that can be smoothly deformed to reticulate a sphere. Many of these cases are achiral; for example among the edge-graphs of the 5 Platonic and 13 Archimedean polyhedra, only the snub cube and the snub dodecahedron are (topologically) chiral. We can however, tangle the network, so that it remains topologically polyhedral, but no longer reticulates a sphere. The simplest entanglements reticulate the donut-shaped torus rather than the sphere. To our surprise, we have failed to generate a single example of a toroidal polyhedral graph that is achiral. We conjecture here that all toroidal entanglements of polyhedral graphs are chiral. The conjecture is proven provided the entanglement contains a knot or a link. For convenience here, we refer to the family of embeddings of a graph that preserve ambient isotopy – viz. all possible realisations of the graph can be deformed into each other without passing edges through each other – as isotopes. For a given graph, there exist an unlimited number of distinct isotopes, partially characterised by the presence of distinct families of knots and links within minors of the graph. (The minor of a graph results when vertices are removed, along with the edges that connect to them, or if edges are deleted, or contracted so that their endpoints merge.) We have recently suggested a ranked enumeration schema for isotopes based on embeddings of the isotope within orientable 2-manifolds of increasing topological complexity. We describe the genus of the isotope as being equal to that of the (topologically) simplest orientable manifold that can be reticulated by the isotope such that no disjoint edges in the isotope cross in the manifold, also referred to as the ‘minimal embedding’ of the isotope. That approach is particularly suited to the enumeration of polyhedral graphs, i.e. 3-connected planar simple graphs. A 3-connected graph is one that can have no fewer than three vertices (and attendant edges) removed before it forms two or more disconnected components. (We note in passing that if the connectivity of the graph is larger than three, it is also 3-connected.) A simple graph is one in which there is a maximum of one edge between any pair of vertices; while a planar graph is one that can be embedded in the plane (or equivalently the sphere) without edge crossings. The simplest – unknotted – embedding of finite polyhedral graphs is in the 2D sphere, S. Higher-order isotopes embed in the genus-one torus (the donut), the genus-two torus (the ‘bitorus’), etc. A preliminary account of this approach has been published recently and a fuller account, dealing with genus-one toroidal embeddings of the tetrahedral, octahedral and cube graphs, is in preparation. Here we focus on embedded graphs that result from reticulations of the torus by polyhedral graphs, embedded in 3-space in the standard manner (as a donut). We consider only those isotopes that cannot be realised by spherical reticulations; we call these ‘toroidal isotopes’ and describe the toroidal reticulation as a ‘minimal embedding’. The tetrahedral graph is the simplest polyhedral graph. Evidently, the standard embedding of the unknotted Dept of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 0200, Australia. E-mail: [email protected] w Electronic supplementary information (ESI) available: Discussion of toroidal reticulation: Fig. 1: Film 1. Selected frames of the computer animation showing the formation of a toroidal isotope (bottom right) from its universal cover (top left). See DOI: 10.1039/b907338h