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Title: Some topics in topological graph theory motivated by chemistry
Author: Barthel, Senja Dominque
ISNI:       0000 0004 6495 5025
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2015
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Topological graph theory is a field of geometric topology. The mathematical objects of interest are embeddings of graphs in 3-space. The image is a so called spatial graphs. A spatial graph can be seen as a generalised knot. In addition to the resulting richer structure, questions about spatial graphs can also be motivated from other natural sciences. In particular, there are many applications to chemistry since molecules can be modelled as graphs embedded in R3. This text consists of two parts. Both cover pure mathematical problems which are motivated by questions from synthetic chemistry. The aim is to find materials with new chemical/physical properties. The structural richness of entangled, catenated and knotted structures has long been a target for synthetic chemistry. The first part investigates the behaviour of entanglements in spatial graphs that are not caused by knotted or linked subgraphs with respect to the surfaces the spatial graphs embed in. We show that all nontrivial embeddings of abstractly planar graphs on the torus contain either a nontrivial knot or a nonsplit link. It follows that ravels do not embed on the torus which was conjectured by Castle, Evans and Hyde in 2008. Our results provide general insight into properties of molecules that are synthesised on a torus. The second part predicts the topologically possible braided structures of 1-dimensional coordination polymers. Given the common way of synthesising via self-assembly, these coordination polymers can be modelled by pure braids with n rigidly congruent strands up to chirality. We discuss the properties and symmetries of 1-dimensional coordination polymers with up to five strands. This project is part of a collaboration with Prof D. M. Proserpio, Dr I. A. Baburin and Dr F. D.-H. Lau.
Supervisor: Buck, Dorothy Sponsor: Imperial College London ; Deutscher Akademischer Austauschdienst ; Evangelisches Studienwerk Villigst
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral