Entanglement of Embedded Graphs

Abstract

We discuss the identification of untangled graph embeddings for finite planar and nonplanar graphs as well as infinite crystallographic nets. Two parallel approaches are discussed: explicit 3-space embeddings and reticulations of 2-manifolds. 2D and 3D energies are proposed that allow ranking of (un)tangled embedding graphs. §1. Introduction Graphs, G are topological objects, composed of a collection of vertices, v i (i ∈ {1, N}) and edges, e j , associated with vertex pairs, e j (v k , v l ). For simplicity, assume that G is simple (i.e. for any {k, l}, the number of associated edges is at most one). We are interested in embeddings of topological graphs in euclidean 3-space, G, motivated by the plethora of three-dimensional chemical structures, which can be idealised as embedded graphs. The vertices and edges of these graphs correspond to atoms and chemical bonds in covalent crystals or organic molecules; in the case of supra-molecular materials such as metal-organic frameworks (MOFs) or DNA assemblies) these coincide with molecular groups and polymeric ligands or H-bonds respectively. We are interested in the effects of the graph embedding on the behaviour of the material.These issues are relevant to the physical properties of materials. For example, the hardness of physical glasses can be correlated with the rigidity in the resulting bonding network, and is therefore critically dependent on the topology of the glass network. 22) The viscosity of polymers in solution is intimately associated with the entanglement of the polymer chains within the solution. 10) Here we explore aspects of ambient isotopy of graph embeddings. Define all embeddings G of a graph G that share a common ambient isotopy as equivalent isotopes. Distinct isotopeswhich are not ambient isotopic -differ in the relative entanglement of graph edges. Definition of entanglement requires elucidation of an unentangled 'ground state' isotope, G 0 , of a graph. We propose a definition for G 0 for generic G, including infinite graphs, later in this paper. For now, we assume G is 3-connected and planar (and simple), so that it is a polyhedral graph. 11) In this case, G 0 necessarily embeds in the sphere (S 2 ) without edge crossings. This is also sufficient to characterise G 0 , since Whitney's theorem ensures that the 2-cell embedding of G into S 2 -the isotope G 0 -is unique. 24) Downloaded from https://academic.oup.com/ptps/article-abstract

Periodic entanglement I: networks from hyperbolic reticulations

“…That approach effectively 'untangles' any net embedding formed on a TPMS, giving an embedding with straight edges (Hyde & Delgado-Friedrichs, 2011;Castle et al, 2011). That approach effectively 'untangles' any net embedding formed on a TPMS, giving an embedding with straight edges (Hyde & Delgado-Friedrichs, 2011;Castle et al, 2011).…”

“…Recall that 'untangled' nets are ambient isotopic to a barycentric embedding (Castle et al, 2011). That is the possibility of entangled nets that are themselves tangled.…”

“…Embeddings that are not ambient isotopic, and have distinct entanglements, are distinct isotopes (Castle et al, 2011). We consider any pair of embeddings to be equivalent if they share the same entanglements (notwithstanding the associated difficulties in characterization of that entanglement).…”

High-symmetry embeddings of interpenetrating periodic nets. Essential rings and patterns of catenation

“…Two embeddings are ambient isotopic if one can be deformed into the other without links passing through each other. Structures with the same topology but not ambient isotopic have been called isotopes (Castle et al, 2011).…”

Ideal geometry of periodic entanglements

“…Its length is equal to that of the ideal torus knot, plus the additional two short connecting edges, which gives a ropelength of 33.22. If we rank entanglement complexity by ideal ropelength, all tangled isotopes are more complex than their untangled isotope, as expected, and the tetrahedron isotopes naturally order in the sequence A, B, C, E, D. Alternatively, we can rank the isotopes via their relaxed length within the flat two-dimensional torus, giving an effective length in their two-dimensional embedding [26]. The resulting two-dimensional energy for these graph embeddings, orders the isotopes as A, B, C, D, E, slightly different to our three-dimensional ropelength ranking (table 3).…”