Thesis advisor: John A. Baldwin / We study the relationship between knots in contact three-manifolds and complex curves in Stein surfaces. To do so, we extend the notion of quasipositivity from classical braids to links that are braided with respect to an open book decomposition of an arbitrary closed, oriented three-manifold. Our main results characterize the transverse links in Stein-fillable contact three-manifolds that bound smooth holomorphic curves in Stein fillings. This characterization is made possible by new techniques in the theory of characteristic and open book foliations on surfaces in three-manifolds. We also explore the Seifert genera of cross-sections of complex plane curves, minimal braid representatives of quasipositive links, and the relationship between Legendrian ribbons in contact three-manifolds and strongly quasipositive braids with respect to compatible open books. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: John A. Baldwin / Contact geometry has played a central role in many recent advances in low-dimensional topology; e.g. in showing that knot Floer homology detects the genus of a knot and whether a knot is fibered. It has also been used to show that the unknot, trefoil, and figure eight knot are determined by their Dehn surgeries. An important problem in 3-dimensional contact geometry is the classification of Legendrian and transverse knots. Such knots come equipped with some classical invariants. New invariants from knot Floer homology have been effective in distinguishing Legendrian and transverse knots with identical classical invariants, a notoriously difficult task. The Giroux correspondence allows contact structures to be studied via purely topological constructs called open book decompositions. Transverse links are then braids about these open books, which in turn may be thought of as mapping tori of diffeomorphisms of compact surfaces with boundary having marked points, which we refer to as pointed monodromies. In the first part of this thesis, we investigate properties of the transverse invariant in knot Floer homology, in particular its behavior for transverse closures of pointed monodromies possessing certain dynamical properties. The binding of an open book sits naturally as a transverse link in the supported contact manifold. We prove that the transverse link invariant in knot Floer homology of the binding union any braid about the open book is non-zero. As an application, we show that any pointed monodromy with fractional Dehn twist coefficient greater than one has non-zero transverse invariant, generalizing a result of Plamenevskaya for braids about the unknot. In the second part of this thesis, we define invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing those defined by Ozsvath, Szabo and Thurston. We show that our invariants are equivalent to those defined by Lisca, Ozsvath, Szabo and Stipsicz for Legendrian and transverse links in arbitrary contact 3-manifolds. Our argument involves considering braids about rational open book decompositions and filtrations on knot Floer complexes. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: Julia E. Grigsby / The first part of this thesis is on properties of annular Khovanov homology. We prove a connection between the Euler characteristic of annular Khovanov homology and the classical Burau representation for closed braids. This yields a straightforward method for distinguishing, in some cases, the annular Khovanov homologies of two closed braids. As a corollary, we obtain the main result of the first project: that annular Khovanov homology is not invariant under a certain type of mutation on closed braids that we call axis-preserving. The second project is joint work with Adam Saltz. Plamenevskaya showed in 2006 that the homology class of a certain distinguished element in Khovanov homology is an invariant of transverse links. In this project we define an annular refinement of this element, kappa, and show that while kappa is not an invariant of transverse links, it is a conjugacy class invariant of braids. We first discuss examples that show that kappa is non-trivial. We then prove applications of kappa relating to braid stabilization and spectral sequences, and we prove that kappa provides a new solution to the word problem in the braid group. Finally, we discuss definitions and properties of kappa in the reduced setting. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Knots and links $L\subset S^3$ carry a wealth of data. Spanning surfaces $F$ (1- or 2-sided), $\partial F=L$, especially {\bf checkerboard} surfaces from link diagrams $D\subset S^2$, help to mine this data. This text explores the structure of these surfaces, with a focus on a gluing operation called {\bf plumbing}, or {\it Murasugi sum}. First, naive classification questions provide natural and accessible motivation for the geometric and algebraic notions of essentiality (incompressibility with $\partial$-incompressibility and $\pi_1$-injectivity, respectively). This opening narrative also scaffolds a system of hyperlinks to the usual background information, which lies out of the way in appendices and glossaries. We then extend both notions of essentiality to define geometric and algebraic {\it degrees} of essentiality, $\underset{\hookrightarrow}{\text{ess}}(F)$ and $\text{ess}(F)$. For the latter, cutting $S^3$ along $F$ and letting $\mathcal{X}$ denote the set of compressing disks for $\partial (S^3\backslash\backslash F)$ in $S^3\backslash\backslash F$, $\text{ess}(F):=\min_{X\in\mathcal{X}}|\partial X\cap L|$. Extending results of Gabai and Ozawa, we prove that plumbing respects degrees of algebraic essentiality, $\text{ess}(F_1*F_2)\geq\min_{i=1,2}\text{ess}(F_i)$, provided $F_1,F_2$ are essential. We also show by example that plumbing does not respect the condition of geometric essentiality. We ask which surfaces de-plumb uniquely. We show that, in general, essentiality is necessary but insufficient, and we give various sufficient conditions. We consider Ozawa's notion of representativity $r(F,L)$, which is defined similarly to $\text{ess}(F)$, except that $F$ is a closed surface in $S^3$ that contains $L$, rather than a surface whose boundary equals $L$. We use Menasco's crossing bubbles to describe a sort of thin position for such a closed surface, relative to a given link diagram, and we prove in the case of alternating links that $r(F,L)\leq2$. (The contents of Chapter 4, under the title Alternating links have representativity 2, are first published in Algebraic \& Geometric Topology in 2018, published by Mathematical Sciences Publishers.) We then adapt these arguments to the context of spanning surfaces, obtaining a simpler proof of a useful crossing band lemma, as well as a foundation for future attempts to better classify the spanning surfaces for a given alternating link. We adapt the operation of plumbing to the context of Khovanov homology. We prove that every homogeneously adequate Kauffman state has enhancements $X^\pm$ in distinct $j$-gradings whose traces (which we define) represent nonzero Khovanov homology classes over $\mathbb{Z}/2\mathbb{Z}$, and that this is also true over $\mathbb{Z}$ when all $A$-blocks' state surfaces are two-sided. A direct proof constructs $X^\pm$ explicitly. An alternate proof, reflecting the theorem's geometric motivation, applies our adapted plumbing operation. Finally, we describe an interpretation of Khovanov homology in terms of decorated cell decompositions of abstract, nonorientable surfaces, featuring properly embedded (1+1)-dimensional nonorientable cobordisms in (2+1)-dimensional nonorientable cobordisms. This formulation contains a planarity condition; removing this condition leads to Khovanov homology for virtual link diagrams.

We begin by introducing knots and links generally and identifying various geometric, polynomial, and integer-based knot and link invariants. Of particular importance to this paper are ternary operations and Niebrzydowski tribrackets defined in [12], [10]. We then introduce multi-tribrackets, ternary algebraic structures following the specified region coloring rules with di↵erent operations at multi-component and single component crossings. We will explore examples of each of the invariants and conclude with remarks on the direction of the introduced multi-tribracket theory.

The traditional skein relation for the Alexander polynomial involves an oriented knot, K+, with a distinguished positive crossing; a knot K−, obtained by changing the distinguished positive crossing of K+ to a negative crossing; and a link K0, the orientation preserving resolution of the distinguished crossing. We refer to (K+,K−,K0) as the oriented skein triple. A tangle is defined as a pair (B, t) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t. DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K− ↔ K+. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K± ↔ K0. The oriented triple is now viewed as K− = circular DNA substrate, K+ = product of topoisomerase action, K0 = product of recombinase action. The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K+ and K−, that differ by a crossing change and a link, K0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins. In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology, was constructed by Peter Ozsváth, Zoltán Szabó, and independently Jacob Rasmussen, denoted by HFK. It is also a refinement of a classical invariant, the Alexander polynomial. The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action. We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel links.

No description available.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.

An evolutionary geometry parametrization is established to represent aerodynamic configurations. This geometry parametrization technique is constructed by integrating the classical B-spline formulation with the knot insertion algorithm. It is capable of inserting control points to a given parametrization without modifying its geometry. Taking advantage of this technique, a shape design problem can be solved as a sequence of optimizations from the basic parametrization to more refined parametrizations. Owing to the nature of the B-spline formulation, feasible parametrization refinements are not unique; guidelines based on sensitivity analysis and geometry constraints are developed to assist the automation of the proposed optimization sequence. Test cases involving airfoil optimization and induced drag minimization are solved adopting this method. Its effectiveness is demonstrated through comparisons with optimizations using uniform refined parametrizations.

An evolutionary geometry parametrization is established to represent aerodynamic configurations. This geometry parametrization technique is constructed by integrating the classical B-spline formulation with the knot insertion algorithm. It is capable of inserting control points to a given parametrization without modifying its geometry. Taking advantage of this technique, a shape design problem can be solved as a sequence of optimizations from the basic parametrization to more refined parametrizations. Owing to the nature of the B-spline formulation, feasible parametrization refinements are not unique; guidelines based on sensitivity analysis and geometry constraints are developed to assist the automation of the proposed optimization sequence. Test cases involving airfoil optimization and induced drag minimization are solved adopting this method. Its effectiveness is demonstrated through comparisons with optimizations using uniform refined parametrizations.