Since its inception, the notion of thin position has played an important role in low-dimensional topology. Thin position for knots in the 3-sphere was first introduced by David Gabai in order to prove the Property R Conjecture. In addition, this theory factored into Cameron Gordon and John Luecke's proof of the knot complement problem and revolutionized the study of Heegaard splittings upon its adaptation by Martin Scharlemann and Abigail Thompson. Let h be a Morse function from the 3-sphere to the real numbers with two critical points. Loosely, thin position of a knot K in the 3-sphere is a particular embedding of K which minimizes the total number of intersections with a maximal collection of regular level sets, where this number of intersections is called the width of the knot. Although not immediately obvious, it has been demonstrated that there is a close relationship between a thin position of a knot K and essential meridional planar surfaces in its exterior E(K). In this thesis, we study the nature of thin position under knot companionship; namely, for several families of knots we establish a lower bound for the width of a satellite knot based on the width of its companion and the wrapping or winding number of its pattern. For one such class of knots, cable knots, in addition to finding thin position for these knots, we establish a criterion under which non-minimal bridge positions of cable knots are stabilized. Finally, we exhibit an embedding of the unknot whose width must be increased before it can be simplified to thin position.

We study the sl(3) web algebra via morphisms on foams. A pre-foam is a cobordism between two webs that contains singular arcs, which are sets of points whose neighborhoods are homeomorphic to the cross-product of the letter "Y'' and the unit interval. Pre-foams may have a distinguished point, and it can be moved around as long as it does not cross a singular arc. A foam is an isotopy class of pre-foams modulo a set of certain relations involving dots on the pre-foams. Composition in Foams is achieved by stacking pre-foams. We compute the cohomology ring of the sl(3) web algebra and apply a functor from the cohomology ring of the sl(3) web algebra to {\bf Foams}. Afterwards, we use this to study the $\mathfrak{sl}(3)$ web algebra via morphisms on foams.

The first part of our work is a statistical and geometric study of properties of Thompson's Group F. We enumerate the number of elements of F which are represented by a reduced pair of n-caret trees, and give asymptotic estimates. We also discuss the effects on word length and number of carets of right multiplication by a standard generator x0 or x1. We enumerate the average number of carets along the left edge of an n-caret tree, and use an Euler transformation to make some conjectures relating to right multiplication by a generator. We describe a computer algorithm which produces Fordham's Table, and discuss using the computer algorithm to find a corresponding Fordham's Table for different generating sets for F. We expound upon the work of Cleary and Taback by completely classifying dead end elements of Thompson's group, and use the classification to discuss the spread of dead end elements and describe interesting elements we call deep roots. We discuss how deep roots may aid in answering the amenability problem for Thompson's group. The second part of our work deals with random facet pairings of simplexes. We show that a random endpoint pairings of segments most often results in a disconnected one-manifold, and relate this to a game called "The Human Knot." When the dimension of the simplexes is greater than 1, however, a random facet pairing most often results in a connected pseudo manifold. This result can be stated in terms of graph theory as follows. Most regular multi graphs are connected, as long as the common valence is at least three.

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Thesis(M.Sc.( Agriculture, Horticulture)) -- University of Limpopo, 2019 / Worldwide, tomato (Solanum lycopersicum L.) is one of the most important crops grown for nutritional value and health benefits, and are highly susceptible to root-knot (Meloidogyne species) nematodes. Following the withdrawal of synthetic chemical nematicides, Nemarioc-AL and Nemafric-BL phytonematicides have been researched and developed as alternatives to synthetic chemical nematicides. However, Nemarioc-AL and Nemafric-BL phytonematicides contains allelochemicals namely, cucurbitacin A (C32H46O9) and cucurbitacin B (C32H46O8) as their active ingredients. Therefore, the objective of this study was to determine whether increasing concentration of Nemarioc AL and Nemafric-BL phytonematicides would result in cucurbitacin residues in tomato plant, to generate mean concentration stimulation point (MCSP) values, overall sensitivity (∑k) and selected foliar mineral elements of tomato plant. Two parallel trials of Nemarioc AL and Nemafric-BL phytonematicides were conducted under field conditions, with each validated the next season. Each trial had seven treatments, namely, 0, 2, 4, 8, 16, 32 and 64% of Nemarioc-AL or Nemafric-BL phytonematicide concentrations, arranged in a randomised complete block design (RCBD), with five replications. In each trial, the seasonal interaction on variables was not significant and therefore data were pooled across the two seasons (n = 70). In both phytonematicides, the cucurbitacin residues were not detected in soil and tomato fruit. Plant variables and selected foliar nutrient elements were subjected to the Curve-fitting Allelochemical Response Data (CARD) model to generate biological indices which allowed for the calculation of MCSP of phytonematicides on tomato and their ∑k values of tomato to Nemarioc-AL and Nemafric BL phytonematicides. In Nemarioc-AL phytonematicide experiment, MCSP for tomato plant variables was at 1.13%, with the ∑k of 60 units, while the MCSP for selected tomato nutrient elements in leaf tissues was at 2.49%, with the ∑k of 21 units. Plant height, chlorophyll content, stem diameter, number of fruit, dry fruit mass, dry shoot mass and dry root mass each with increasing concentration of Nemarioc-AL phytonematicide exhibited positive quadratic relations with a model explained by 95, 82, 96, 89, 83, 83 and 92%, respectively. Similarly, K, Na and Zn each with increasing Nemarioc-AL phytonematicide concentration exhibited positive quadratic relations with a model explaining a strong relationship by 91, 96 and 89%. In Nemafric-BL phytonematicide experiment, MSCP for tomato plant variables was at 1.75%, with the ∑k of 45 units, whereas MCSP for selected tomato nutrient elements in leaf tissues was at 3.72% with the ∑k of 33 units. Plant height, chlorophyll content, stem diameter, number of fruit, dry fruit mass, dry shoot mass and dry root mass and increasing Nemafric-BL phytonematicide concentration exhibited positive quadratic relations with the model explaining a strong relationship by 92, 83, 97, 96, 87, 94 and 96%. Likewise, Na and Zn each with increasing Nemafric-BL phytonematicide concentration exhibited positive quadratic relations with a model explaining their relationship by 93 and 83%, respectively. In contrast, K with increasing Nemafric-BL phytonematicide concentration exhibited negative quadratic relations with a model explaining the relationship by 96%. In conclusion, tomato plant variables and selected foliar nutrient elements over increasing concentration of phytonematicides exhibited DDG patterns, characterised by three phases, namely, stimulation, neutral and inhibition. The developed non-phytotoxic concentration would be suitable for successful tomato production under field conditions.

Knot theory has rapidly expanded in recent years. New representations of braid groups led to an extremely powerful polynomial invariant, the Jones polynomial. Combinatorics applied to knot and link diagrams led to generalizations. Knot theory also has connections with other fields such as statistical mechanics and quantum field theory, and has applications in determining how certain enzymes act on DNA molecules, for example. The principal objective of this dissertation is to study the relations between knots and algebraic structures called quandles. A quandle is a set with a binary operation satisfying some properties related to the three Reidemeister moves. The study of quandles in relation to knot theory was intitiated by Joyce and Matveev. Later, racks and their (co)homology theory were defined by Fenn and Rourke. The rack (co)homology was also studied by Grana from the viewpoint of Hopf algebras. Furthermore, a modified definition of homology theory for quandles was introduced by Carter, Jelsovsky, Kamada, Langford, and Saito to define state-sum invariants for knots and knotted surfaces, called quandle cocycle invariants. This dissertation studies the quandle cocycle invariants using extensions of quandles and knot colorings. We obtain a coloring of a knot by assigning elements of a quandle to the arcs of the knot diagram. Such colorings are used to define knot invariants by state-sum. For a given coloring, a 2-cocycle is assigned at each crossing as the Boltzmann weight. The product of the weights over all crossings is the contribution to the state-sum, which is the formal summation of the contributions over all possible colorings of the given knot diagram by a given quandle. Generalizing the cocycle invariant for knots to links, we define two kinds of invariants for links: a component-wise invariant, and an invariant defined as families of vectors. Abelian extensions of quandles are also defined and studied. We give a formula for creating infinite families of abelian extensions of Alexander quandles. These extensions give rise to explicit formulas for computing 2-cocycles. The theory of quandle extensions parallels that of groups. Moreover, we investigate the notion of extending colorings of knots using quandle extensions. In particular, we show how the obstruction to extending the coloring contributes to the non-trivial terms of the cocycle invariants for knots and links. Moreover, we demonstrate the relation between these new cocycle invariants and Alexander matrices.

The diploma thesis called "The Woolgathering" is connected with the idea of gather a wave. It is the image of vanishing moment, effort for hold something that escapes us. Practical part is represented by installation of text, image and audio based on experience, reflection and reverie.

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The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every nonsingular, prime alternating link and all torus links, and explore some of their properties and applications. Several examples are exhibited with color coding of tiles.