Characterisation of Spherical Splits

Author: McCone, Thomas Andrew

Date: 2020

Publisher: University of Otago

Type: Thesis

Link to this item using this URL: http://hdl.handle.net/10523/9988

University of Otago

Abstract

We 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.

Subjects: split collections, combinatorics, convex geometry, graph theory, oriented matroids, polytopes

Citation: ["McCone, T. A. (2020). Characterisation of Spherical Splits (Thesis, Master of Science). University of Otago. Retrieved from http://hdl.handle.net/10523/9988"]

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