The hyperbolic volume of a knot complement is a topological knot invariant. Futer, Kalfagianni, and Purcell have estimated the volumes of Montesinos link complements for Montesinos links with at least three positive tangles. Here we extend their results to all hyperbolic Montesinos links.

Thesis advisor: Julia Elisenda Grigsby / In 1999, Khovanov constructed a combinatorial categorification of the Jones polynomial. Since then there has been a question of to what extent the topology of a link is reflected in his homology theory and how Khovanov homology can be used for topological applications. This dissertation compiles some of the authors contributions to these avenues of mathematical inquiry. In the first chapter, we prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Focusing on the case of 3-braids, we compute the Rasmussen $s$-invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $\psi$ invariant is entirely determined by the $s$-invariant and the self-linking number for 3-braid closures. In the second chapter, we show if $L$ is any link in $S^3$ whose Khovanov homology is isomorphic to the Khovanov homology of $T(2,6)$ then $L$ is isotopic to $T(2,6)$. We show this for unreduced Khovanov homology with $\mathbb{Z}$ coefficients. Finally in the third chapter, we exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Virtual knot theory is an extension of classical knot theory based on a combinatorial presentation of crossing information. The appropriate extensions of braid groups and string link monoids have also been studied. While some previously known knot invariants can be evaluated for virtual objects, entirely new techniques can also be used, for example, the concept of index of a crossing, and its resulting (Gaussian) parity theory. In general, a parity is a rule which assigns 0 or 1 to each crossing in a knot or link diagram. Recently, they have also been defined for virtual braids. Here, novel parities for knots, braids, and string links are defined, some of their applications are explored, most notably, defining a new subgroup of the virtual braid groups. / Thesis / Master of Science (MSc)

A template is a key tool that we use to study knotted periodic orbits in the three-dimensional flow. The simplest type of template is the Lorenz template. In [5], Birman and Williams studied knotted periodic orbits with the aid of the Lorenz template. They discovered remarkable properties of Lorenz knots and links. No half twists exist in the Lorenz template. The new template is referred to be a Lorenz-like template when we add half twists. We looked at the template L(1,-1) in this paper, which has a positive half twist on the left-side and a negative half twist on the right. We look for the different types of knots and links that the template contains. Afterward, it was discovered that some knot types in L(1,-1) are fibered. Additionally, we look into the linking number of links in L(1,-1), as well as L(m; n) for m > 0 and n < 0. We have also explored the subtemplate of L(1,-1).

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No description available.

A stochastic framework to simulate the process of initiation, diameter growth, death and self-pruning of branches in loblolly pine (Pinus taeda L.) trees was developed. A data set was obtained from a destructive sampling of whorl sections from 34 trees growing under different initial spacing. Data from dissected branches were used to develop a model for representing knot shape, which assumed that the live portion of a knot can be modeled by a one-parameter equation and the dead portion by assuming a cylindrical shape. For the developed knot model analytical expressions were derived for estimating the volume of knots (live/dead portions) for three types of branch conditions on simulated trees: (i) live branches, (ii) non-occluded dead branches, and (iii) occluded dead branches. This model was intended to recover information on knots shape and volume during the simulation process of branch dynamics. Three different components were modeled and hierarchically connected: whorl, branches and knots. For each new growing season, whorls and branches are assigned stochastically along and around the stem. Thereafter, branch diameter growth is predicted as function of relative location within the live crown and stem growth. Using a taper equation, the spatial location (X,Y,Z) of both live and dead portion of simulated knots is maintained in order to create a 3D representation of the internal stem structure. At the end of the projection period information on (i) vertical trend of branch diameter and location along and around the stem, (ii) volume of knots, and (iii) spatial location, size and type (live and dead) of knots can be obtained. The proposed branch model was linked to the individual-tree growth and yield model PTAEDA3.1 to evaluate the effect of initial spacing and thinning intensity on branch growth in sawtimber trees. The use of the dynamic branch model permitted generation of additional information on sawlog quality under different management regimes. The arithmetic mean diameter of the largest four branches, one from each radial quadrant of the log (i.e. Branch Index, BI) and the number of whorls per log were considered as indicators of sawlog quality. The developed framework makes it possible to include additional wood properties in the simulation system, allowing linkage with industrial conversion processes (e.g. sawing simulation). This integrated modeling system should promote further research to obtain necessary data on crown and branch dynamics to validate the overall performance of the proposed branch model and to improve its components. / Ph. D.

A study was undertaken to investigate the morphological host-parasite interactions of selected resistant and susceptible lines of tomato to Meloidogyne incognita, M. incognita var. acrita, M. javanica, M. hapla and M. arenaria and to determine the mode of inheritance of nematode resistance and the number of genetic factors controlling resistance to the root-knot nematodes. Four resistant varieties of tomato were crossed with one susceptible variety. The F₁ populations showed hybrid vigor for height, yield, and fresh weight of roots, stems, and leaves. Resistance to M. javanica, M. incognita, M. incognita var. acrita was dominant and susceptibility was recessive. The F₂ populations segregated in a 3:1 ratio showing resistance is a monofactorial dominant character and controlled by the same gene. The resistant parents, and the F₁ and F₂ populations did not show resistance to M. hapla and M. arenaria. Anatomical studies showed that there were some slight differences between resistant and susceptible varieties. In resistant varieties a compact layer of cells was formed around the body of the nematode which may have caused the noticeable reduction in nematode development and egg-formation. Giant cells formed in resistant varieties were much smaller and fewer in number than in susceptible varieties. The contrast between these two reactions by the resistant and the susceptible hosts suggests that resistance is related to the cellular response of the host to the parasite. In the root penetration and attraction study it was observed that when 2000 larvae were used as inoculum, they freely penetrated the roots of susceptible seedlings whereas in resistant roots very few larvae entered and most remained half embedded in the roots even at the 96 hour interval after inoculation. When the concentration of the larvae inoculum was increased from 2000 to 8000 per seedling, the larvae entered the roots of resistant seedlings as freely and as rapidly as they entered the roots of susceptible ones, demonstrating that the concentration of inoculum is an important factor in penetration. / Ph. D.

This is a study in which landscape architecture is theoretically related to the &quot;textile art.&quot; It establishes a theoretical analogy of the landscape as a kind of textual manifestation, &quot;the landscape is a textile,&quot; and aims to establish new resemblances that show how the landscape and textile arts are related, not only with regards to the elements of composition, or to similarities between the elemental relationships that exist in both these arts, but to how the study of structure and form in the production of textiles may influence our understanding of the textile nature of the landscape. The first part of the research is developing a theoretical analogy between landscape and fabric. The process of making textiles is based on weaving and knitting, operations in which knots obviously play a most important role. The context of the urban landscape can also be viewed as a woven fabric of different threads, where knots are the summit of this interwoven textile. This study shows that the goal of landscape is to knit together the clusters of meaning so that the person can experience the unity that binds up these different qualities. Based on this theoretical analogy, the second part uses the &quot;action research&quot; method which in the context of this study would be a scholarly practice of design, &quot;design-research.&quot; Both parts of the research are qualitative inquiry in nature and the qualitative manner of the investigation calls for an inductive investigation rather than a deductive one; theoretical discussions and the design section rely heavily on interpretation of the researcher. / Master of Landscape Architecture

We developed and tested a novel measure of non-rigid, ductile spatial skill using knot reasoning. In Study 1, 279 US adults (M = 30.90 SD = 5.47 years; 76% White, 48% women) recruited through Prolific completed a 73-item knot reasoning task. Using Item Response Theory, we tested the reliability of the measure and removed items with low discrimination to yield a final 54-item measure with good reliability (α = .88). In Study 2, 147 US adults (M = 20.65 SD = 2.80 years; 48% White, 56% women) recruited from a public university in the mid-Atlantic completed a battery of existing spatial skills measures, the new knot reasoning measure, a control verbal measure, and a survey of current and childhood spatial activities. We validated the knot reasoning measure: performance on this measure is significantly, positively correlated with existing measures of spatial skill (mental rotation, paper folding, bending). However, we did not find support for a continuum of spatial skills from non-rigid to rigid using a simultaneous regression and a confirmatory factor analytic approach. Finally, we replicated prior work showing a male advantage in mental rotation performance but no gender differences in other spatial skills, though this relationship differed when using a modeling approach that incorporated spatial activities experience. Using a structural equation modeling approach, we found that masculine-stereotyped spatial activities engagement mediated the relationship between gender and mental rotation and knot reasoning task performance, where men who reported fewer spatial activities had higher spatial skills. Current and childhood feminine-stereotyped spatial engagement mediated the relationship between gender and paper folding performance, with women who reported greater spatial activities had higher spatial skills. Finally, we found that spatial skills did not differ among math-intensive STEM, non-math-intensive STEM, and non-STEM majors. / Psychology