Entangled graphs on surfaces in space
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In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory...[Show more]
dc.contributor.author  Castle, Toen  

dc.date.accessioned  20140827T23:38:28Z  
dc.date.available  20140827T23:38:28Z  
dc.identifier.other  b35683934  
dc.identifier.uri  http://hdl.handle.net/1885/11978  
dc.description.abstract  In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a threedimensional problem to a twodimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical selfassembly, including DNA nanotechnology and metalligand complex crystallisation.  
dc.language.iso  en_AU  
dc.subject  geometry  
dc.subject  topology  
dc.subject  entanglement  
dc.subject  chirality  
dc.subject  knots  
dc.subject  links  
dc.subject  ravels  
dc.subject  platonic polyhedra  
dc.subject  cube  
dc.subject  tetrahedron  
dc.subject  octahedron  
dc.subject  toroidal embedding  
dc.title  Entangled graphs on surfaces in space  
dc.type  Thesis (PhD)  
local.contributor.supervisor  Hyde, Stephen  
local.contributor.supervisorcontact  stephen.hyde@anu.edu.au  
local.type.degree  Doctor of Philosophy (PhD)  
dc.date.issued  2013  
local.contributor.affiliation  Research School of Physics, Department of Mathematics  
local.identifier.doi  10.25911/5d7391e72cdc9  
local.mintdoi  mint  
Collections  Open Access Theses 
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