The notation of t/i numbers is used to describe knots and links on the double torus in two different ways, as a step toward the eventual classification of double-torus links.

Thesis (M. Agric. (Plant Production)) -- University of Limpopo, 2019 / Phytonematicides have allelochemicals as active ingredients and could be highly phytotoxic on crops being protected against nematode damage. In order to avoid phytotoxicity, the application concentration, technically referred to as mean concentration stimulation point (MCSP), along with the application interval, have to be empirically established. The Curve-fitting Allelochemical Response Data (CARD) computer-based model was adopted at the Green Biotechnologies Research Centre of Excellence (GBRCE) for developing the MCSP. The MCSP is computed from the CARD-generated biological indices and was technically defined as a phytonematicide concentration that could manage the nematode population densities without causing phytotoxicity to the test crop and it is plant-specific. The MCSP and application interval had been empirically established for different crops, but they had not been established for sweet potatoes. Therefore, the objective of the study was to determine the MCSP for Nemarioc-AL phytonematicide on Meloidogyne javanica-infected sweet potato cv. ꞌBopheloꞌ and its application interval. Sweet potato cuttings were planted in 25-cm diameter plastic bags containing steam-pasteurised loam soil and Hygromix at 3:1 (v/v) ratio. Each plant was inoculated with 5 000 eggs and second-stage juveniles (J2) of M. javanica, with seven treatments, namely, 0, 2, 4, 8, 16, 32 and 64% Nemarioc-AL phytonematicide, arranged in a randomised complete block design, with five replicates. At 56 days after the initiation of treatment, the MCSP values for plant variables and plant physiology variables were 1.92 and 3.08% Nemarioc-AL phytonematicide, respectively. The overall sensitivity values for plant variables and plant physiology variables were 0 and 1 unit, respectively, showing that the sweet potato cv. ꞌBopheloꞌ was highly sensitive to the product. Nematode variables with increasing concentrations of Nemarioc-AL phytonematicide exhibited positive and quadratic relations. The life cycle of M. javanica and the derived MCSP were used to empirically establish the application interval. Briefly, the location and most materials and methods were as outlined above except that ‘weeks-per-month-of-30 days’, with the MCSP being applied on 0, 7.5, 15, 22.5 and 30 days (0, 1, 2, 3 and 4 weeks) serving as treatments, replicated eight times. At 56 days after the treatments, plant variables and increasing application interval exhibited positive quadratic relations with the average of 2.55 ‘week of-30-day-month’ translating to 19 days (2.55/4 × 30), with nematode variables exhibiting negative quadratic relationships. In conclusion, when the MCSP of Nemarioc AL phytonematicide on sweet potato cv. 'Bophelo' at 1.92% was applied every 19 days, it would not be phytotoxic, but it would be able to suppress nematode population densities of M. javanica. The MCSP for essential nutrient elements could be reduced to that of plant growth variables, since the products are not intended for use as fertilisers.

The motivation for this thesis is the computer-assisted calculation of the Jones poly- nomial from braid words in the Artin braid group on three strands, denoted B3. The method used for calculation of the Jones polynomial is the original method that was created when the Jones polynomial was first discovered by Vaughan Jones in 1984. This method utilizes the Temperley-Lieb algebra, and in our case the Temperley-Lieb Algebra on three strands, denoted A3, thus generalizations about A3 that assist with the process of calculation are pursued.

Knot theory is a rich topic in topology that studies the how circles can be embedded in Euclidean 3-space. One of the main questions in knot theory is how to distinguish between different types of knots efficiently. One way to approach this problem is to study knot invariants, which are properties of knots that do not change under a standard set of deformations. We give a brief overview of basic knot theory, and examine a specific knot invariant known as Khovanov homology. Khovanov homology is a homological invariant that refines the Jones polynomial, another knot invariant that assigns a Laurent polynomial to a knot. Dror Bar-Natan wrote a paper in 2002 that explains the construction of Khovanov homology and proves that it is an invariant. We follow his lead and attempt to clarify and explain his formulation in more precise detail.

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Thesis advisor: Joshua E. Greene / We study ribbon cobordisms between 3-manifolds, i.e. rational homology cobordisms that admit a handle decomposition without 3-handles. We first define and study the more general notion of quasi-ribbon cobordisms, and analyze how lattice-theoretic methods may be used to obstruct the existence of a quasi-ribbon cobordism between two given 3-manifolds. Building on this and on previous work of Lisca, we then determine when there exists such a cobordism between two connected sums of lens spaces. In particular, we show that if an oriented rational homology sphere Y admitsa quasi-ribbon cobordism to a lens space, then Y must be homeomorphic to L(n, 1), up to orientation-reversal. As an application, we classify ribbon χ-concordances between connected sums of 2-bridge links. Lastly, we show that the notion of ribbon rational homology cobordisms yields a partial order on the set consisting of aspherical 3-manifolds and lens spaces, thus providing evidence towards a conjecture formulated by Daemi, Lidman, Vela-Vick and Wong. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Embedded contact homology is an invariant of closed oriented contact 3-manifolds first defined by Hutchings, and is isomorphic to both Heegard Floer homology (by the work of Colin, Ghiggini and Honda) and Seiberg-Witten Floer cohomology (by the work of Taubes). The embedded contact chain complex is defined by counting closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the manifold. As part of their proof that ECH=HF, Colin, Ghiggini and Honda showed that if the contact form is suitably adapted to an open book decomposition of the manifold, then embedded contact homology can be computed by considering only orbits and differentials in the complement of the binding of the open book; this fact was then in turn used to define a knot version of embedded contact homology, denoted ECK, where the (null-homologous) knot in question is given by the binding. In this thesis we start by generalizing these results to the case of rational open book decompositions, allowing us to define ECK for rationally null-homologous knots. In its most general form this is a bi-filtered chain complex whose homology yields ECH of the closed manifold. There is also a hat version of ECK in this situation which is equipped with an Alexander grading equivalent to that in the Heegaard Floer setting, categorifies the Alexander polynomial, and is conjecturally isomorphic to the hat version of knot Floer homology. The main result of this thesis is a large negative $n$-surgery formula for ECK. Namely, we start with an (integral) open book decomposition of a manifold with binding $K$ and compute, for all $n$ greater than or equal to twice the genus of $K$, ECK of the knot $K(-n)$ obtained by performing ($-n$)-surgery on $K$. This formula agrees with Hedden's large $n$-surgery formula for HFK, providing supporting evidence towards the conjectured equivalence between the two theories. Along we the way, we also prove that ECK is, in many cases, independent of the choices made to define it, namely the almost complex structure on the symplectization and the homotopy type of the contact form. We also prove that, in the case of integral open book decompositions, the hat version of ECK is supported in Alexander gradings less than or equal to twice the genus of the knot.

Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots}, and flat virtual knots is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We classify flat virtual tangles with no closed components and give bases for its ``infinitesimal'' algebras. The classification of the former can be used as an invariant on virtual tangles with no closed components and virtual braids. In a subsequent paper, we will show that the infinitesimal algebras are the target spaces of any universal finite-type invariants on the respective variants of the flat virtual tangles.

This thesis will take a look at a branch of topology called knot theory. We will first look at what started the study of this field, classical knot theory. Knot invariants such as the Bracket polynomial and the Jones polynomial will be introduced and studied. We will then explore racks and biracks along with the axioms obtained from the Reidemeister moves. We will then move on to generalize classical knot theory to what is now known as virtual knot theory which was first introduced by Louis Kauffman. Finally, we take a look at a newer aspect of knot theory, twisted virtual knot theory and we defined new link invariants for twisted virtual biracks.

Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots}, and flat virtual knots is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We classify flat virtual tangles with no closed components and give bases for its ``infinitesimal'' algebras. The classification of the former can be used as an invariant on virtual tangles with no closed components and virtual braids. In a subsequent paper, we will show that the infinitesimal algebras are the target spaces of any universal finite-type invariants on the respective variants of the flat virtual tangles.