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dc.contributor.advisorBryant, David
dc.contributor.authorMcCone, Thomas Andrew
dc.date.available2020-03-24T03:04:48Z
dc.date.copyright2020
dc.identifier.citationMcCone, T. A. (2020). Characterisation of Spherical Splits (Thesis, Master of Science). University of Otago. Retrieved from http://hdl.handle.net/10523/9988en
dc.identifier.urihttp://hdl.handle.net/10523/9988
dc.description.abstractWe investigate the properties of collections of linear bipartitions of points embedded into $\R^3$, which we call collections of affine splits. Our main concern is characterising the collections generated when the points are embedded into $S^2$; that is, when the collection of splits is spherical. We find that maximal systems of splits occur for points embedded in general position or general position in $S^2$ for affine and spherical splits, respectively. Furthermore, we explore the connection of such systems with oriented matroids and show that a maximal collection of spherical splits map to the topes of a uniform, acyclic oriented matroid of rank 4, which is a uniform matroid polytope. Additionally, we introduce the graphs associated with collections of splits and show that maximal collections of spherical splits induce maximal planar graphs and, hence, the simplicial 3-polytopes. Finally, we introduce some methodologies for generating either the hyperplanes corresponding to a split system on an arbitrary embedding of points through a linear programming approach or generating the polytope given an abstract system of splits by utilising the properties of matroid polytopes. Establishing a solid theory for understanding spherical split systems provides a basis for not only combinatorial–geometric investigations, but also the development of bioinformatic tools for investigating non-tree-like evolutionary histories in a three-dimensional manner.
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherUniversity of Otago
dc.rightsAll items in OUR Archive are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.
dc.subjectsplit collections
dc.subjectcombinatorics
dc.subjectconvex geometry
dc.subjectgraph theory
dc.subjectoriented matroids
dc.subjectpolytopes
dc.titleCharacterisation of Spherical Splits
dc.typeThesis
dc.date.updated2020-03-23T17:02:01Z
dc.language.rfc3066en
thesis.degree.disciplineMathematics and Statistics
thesis.degree.nameMaster of Science
thesis.degree.grantorUniversity of Otago
thesis.degree.levelMasters
otago.openaccessOpen
otago.evidence.presentYes
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