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Recent advances in the computer graphics of woven images on surfaces in 3-space motivate the development of weavings for arbitrary genus surfaces. We present herein a general framework for weaving structures on general surfaces in 3-space, and through it, we demonstrate how weavings on such surfaces are inducible from connected graph imbeddings on the same surfaces. The necessary and sufficient conditions to identify the inducible weavings in our framework are also given. For low genus surfaces, like plane and torus, we extend our framework to the weavings which are inducible from disconnected imbedded graphs. In particular, we show all weavings on a plane are inducible in our framework, including most Celtic Knots. Moreover, we study different weaving structures on general surfaces in 3-space based on our framework. We show that any weaving inducible in our framework can be converted into an alternating weaving by appropriately changing the strand orders at some crossings. By applying a topological surgery operation, called doubling operation, we can refine a weaving or convert certain non-twillable weavings into twillable weavings on the same surfaces. Interestingly, two important subdivision algorithms on graphs imbeddings, the Catmull-Clark and Doo-Sabin algorithms, correspond nicely to our doubling operation on induced weavings. Another technique we used in studying weaving structures is repetitive patterns. A weaving that can be converted into a twillable weaving by our doubling operation has a highly-symmetric structure, which consists of only two repetitive patterns. An extension of the symmetric structure leads to Quad-Pattern Coverable meshes, which can be seamlessly covered with only one periodic pattern. Both of these two topological structures can be represented with simple Permutation Voltage graphs. A considerable advantage of our model is that it is topological. This permits the graphic designer to superimpose strand colors and geometric attributes — distances, angles, and curvatures — that conform to manufacturing or artistic criteria. We also give a software example for plane weaving construction. A benefit of the software is that it supports plane weaving reconstructions from an image of a plane weaving, which could be useful for recording and modifying existing weavings in real life.

<p>The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders.</p>

Any Piecewise-Linear (PL) surface can be formed from a regular polygon (including the interior) with an even number of edges, where the edges are identified in pairs to form a two-dimensional manifold. The resulting surfaces can be distinguished by algebraic means. An analysis of the construction algorithm can also be used to determine the resulting surface. Knowledge of the polygon used can also yield information about the surfaces formed.In this thesis, an algorithm is developed that will analyze all possible edge pairings for an arbitrary regular polygon. The combination of this data, along with known techniques from geometric topology, will categorize the constructions of these PL surfaces. A procedure using matrices is developed that will determine the Euler number and establish which algebraic words are equivalent.This topic extends to two-dimensional manifolds a classical method of analysis for three-dimensional manifolds. It therefore provides a more geometrical approach than has traditionally been used for two dimensional surfaces. / Department of Mathematical Sciences

Black Tracking a Landscape Topology: Extension to Boat Harbour is a path of personal experience and design utilizing both lived and geometric spaces of a design process. The practicum explores and applies landscape topology as a hermeneutic phenomenological approach for designing points along a rejuvenated rail corridor for visitors to engage with a cultural landscape. The Black Track design consists of a path that transects twenty six kilometres through a unique coastal landscape with nine specifically located and designed places intersecting with the layers of natural ecology, past industrial coal mining, and First Nations cultures. This project gathers exposed site elements and celebrates the spirit of place for a design that rehabilitates the rail-bed for a heritage trail experience. The trail reveals the unique cross section of this heritage landscape with subtle signals for experiential discovery design interventions that engage the dimensions of perception and place.

Knot theory is an important branch of mathematics with applications in other branches of science. In this paper, we explore invariants on a special class of knots, known as virtual knots. We find new invariants by taking quotients of quandles, and introducing the fundamental Latin Alexander quandle and its Grobner basis. We also demonstrate examples of computations of these invariants.

This thesis considers the registration of shapes, estimation of shape variability and the statistical modelling of human brain magnetic resonance images (MRI). Current shape registration techniques, such as Procrustes analysis, superimpose shapes in order to make inferences regarding the mean shape and shape variability. We apply Procrustes analysis to a subset of the landmarks and give distributional results for the Euclidean distance of a shape from a template. Procrustes analysis is then generalised to minimise a Mahalanobis norm, with respect to a symmetric, positive denite matrix, and the weighted Procrustes estimators for scaling, rotation and translation obtained. This weighted registration criterion is shown, through a simulation study, to reduce the bias and error in maximum likelihood estimates of the mean shape and covariance matrix compared to isotropic Procrustes. A Bayesian Markov chain Monte Carlo algorithm is also presented and shown to be less sensitive to prior information. We consider two MRI data sets in detail. We examine the first data set for large-scale shape dierences between two volunteer groups, healthy controls and schizophrenia patients. The images are registered to a template through modelling the voxel values and we maximise the likelihood over the transformation parameters. Using a suitable labelling and principal components analysis we show schizophrenia patients have less brain asymmetry than healthy controls. The second data set is a sequence of functional MRI scans of an individual's motor cortex taken while they repeatedly press a button. We construct a model with temporal correlations to estimate the trial-to-trial variability in the haemodynamic response using the Expectation-Maximisation algorithm. The response is shown to change with task and through time. For both data sets we compare our techniques with existing software packages and improvements to data pre-processing are suggested. We conclude by discussing potential areas for future research.

We analyse the perturbative expansion of knot invariants related with infinite dimensional representations of sl(2,R) and the Lorentz group taking as a starting point the Kontsevich Integral and the notion of central characters of infinite dimensional unitary representations of Lie Groups. The prime aim is to define C-valued knot invariants. This yields a family of C([h])-valued knot invariants contained in the Melvin-Morton expansion of the Coloured Jones Polynomial. It is verified that for some knots, namely torus knots, the power series obtained have a zero radius of convergence, and therefore we analyse the possibility of obtaining analytic functions of which these power series are asymptotic expansions by means of Borel re-summation. This process is complete for torus knots, and a partial answer is presented in the general case, which gives an upper bound on the growth of the coefficients of the Melvin-Morton expansion of the Coloured Jones Polynomial. In the Lorentz group case, this perturbative approach is proved to coincide with the algebraic and combinatorial approach for knot invariants defined out of the formal R-matrix and formal ribbon elements in the Quantum Lorentz Group, and its infinite dimensional unitary representations.