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.

This thesis concerns flat knots and their properties. We study various invariants of flat knots, such as the crossing number, the u-polynomial, the flat arrow polynomial, the flat Jones-Krushkal polynomial, the based matrices, and the φ-invariant. We also examine the behavior of these invariants under connected sum and cabling. We give a matrix-based algorithm to calculate the flat Jones-Krushkal polynomial. We take a special interest in certain subclasses of flat knots, such as almost classical flat knots, checkerboard colorable flat knots, and slice flat knots. We explore how the invariants can be used to obstruct a flat knot from being almost classical, checkerboard colorable, or slice. We show that any minimal crossing diagram of a composite flat knot is a con- nected sum, and we introduce a skein formula for the constant term of the flat arrow polynomial. A companion project to this thesis is the interactive website, FlatKnotInfo. It provides a curated dataset of examples and invariants of flat knots. It also features a tool for searching flat knots and another tool that crossreferences flat knots with virtual knots. FlatKnotInfo was used to develop many of the results in this thesis, and we hope others find it useful for their research on flat knots. The Python code for calculating based matrices and flat Jones-Krushkal polynomials is included in an appendix. / Dissertation / Doctor of Philosophy (PhD)

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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.

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.

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.

The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define the Conway polynomial, the Axioms for Computation are given and many examples of their use are included. A major result of this investigation is the proof of topological invariance of these polynomials and the proof that the axioms are sufficient for the calculation of the knot polynomial for any given knot or link.