A knot is an embedding of S1 in three-dimensional space. Generally, it can be thought of as a knotted piece of string with the ends glued together. When we project a knot into the plane, we can create a knot diagram in which we specify which portion of the string lies on top at each place that the string crosses itself. To perform a crossing change on a knot, one can imagine cutting one portion of the string at a crossing, allowing another portion of the string to pass through, and then gluing the cleaved ends back together. We define the distance between two knots, K1 and K2, to be the minimum number of crossing changes one must perform on either K1 or K2 to obtain the other knot. Circular DNA can become knotted during biological processes such as recombination and replication. We can model knotted DNA with a mathematical knot. Type II topoisomerases are the enzymes tasked with keeping DNA unknotted, and they act on double-stranded circular DNA by breaking the backbone of the DNA, allowing another segment of DNA to pass through, and then re-sealing the break. Thus, performing a crossing change on a knot models the action of this protein. Specifically, studying knots of distance one can help us better understand how the action of a type II topisomerase on double-stranded circular DNA can alter DNA topology. We create a knot distance graph by letting the set of vertices be rational knots with up to and including thirteen crossings and by placing an edge between two vertices if the two knots corresponding to those vertices are of distance one. A neighborhood of a vertex, v, in a graph is the set of vertices with which v is adjacent via an edge. Using graph theoretical and topological tools, we examine graphs of knot distances and define a mapping between distance one neighborhoods. Additionally, this idea can also be examined and visualized as performing Dehn surgery on the double branched cover of a knot.

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.

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.

The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.

In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.

Includes vita. / Title from PDF of title page. / Document formatted into pages; contains 81 pages / Thesis (Ph.D.)--University of South Florida, 2002. / Includes bibliographical references. / Text (Electronic thesis) in PDF format. / ABSTRACT: 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. / ABSTRACT: 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. / ABSTRACT: 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. / System requirements: World Wide Web browser and PDF reader. / Mode of access: World Wide Web.

This thesis provides a partial classification of all 3-stranded pretzel knots K=P(p,q,r) with unknotting number one. Scharlemann-Thompson, and independently Kobayashi, have completely classified those knots with unknotting number one when p, q, and r are all odd. In the case where p=2m, we use the signature obstruction to greatly limit the number of 3-stranded pretzel knots which may have unknotting number one. In Chapter 3 we use Greene's strengthening of Donaldson's Diagonalization theorem to determine precisely which pretzel knots of the form P(2m,k,-k-2) have unknotting number one, where m is an integer, m>0, and k>0, k odd. In Chapter 4 we use Donaldson's Diagonalization theorem as well as an unknotting obstruction due to Ozsv\'ath and Szab\'o to partially classify which pretzel knots P(2,k,-k) have unknotting number one, where k>0, odd. The Ozsv\'ath-Szab\'o obstruction is a consequence of Heegaard Floer homology. Finally in Chapter 5 we explain why the techniques used in this paper cannot be used on the remaining cases. / text

This report conveys some of the modern investigations surrounding the use of topology in a contextual setting. Topics discussed include applications of topology relating to the modeling of biological structures and common objects like sunshades, elementary knot theory, and the connection between the fields of topology and algebra. A brief overview and discussion of the incorporation of elementary topology into the standard Geometry curriculum of secondary schools is also examined. / text

In this dissertation we prove that if an n-stranded pretzel knot K has an essential Conway sphere, then there exists an Alexander grading s such that the rank of knot Floer homology in this grading, [mathematical equation], is at least two. As a consequence, we are able to easily classify pretzel knots admitting L-space surgeries. We conjecture that this phenomenon occurs more generally for any knot in S³ with an essential Conway sphere. We also exhibit an infinite family of knots, each of which admits a nontrivial genus two mutant which shares the same total dimension of knot Floer homology, while being distinguished by knot Floer homology as a bigraded invariant. Additionally, the genus two mutation interchanges the [mathematical symbol]-graded knot Floer homology groups in [mathematical symbol]-gradings k and -k. This infinite family of examples supports a second conjecture, namely that the total rank of knot Floer homology is invariant under genus two mutation. / text

The foundation of Luecke’s course M: 396 Geometry and Topology is that collaboration amongst mathematicians and biologists caused tremendous gains in DNA research. The field of topology has led to significant strides in understanding of the topological properties of the genetic molecule DNA. Through the integration of biological phenomena and knowledge of topology and Euclidean geometry, biologists can describe and quantize enzyme mechanisms and therefore determine enzyme mechanisms causing the changes. Understanding mathematical applications in contexts outside of mathematics on any level helps to explain why mathematics is a core content area in primary and secondary education. Requiring secondary educators to take such a course could result in mathematics taught with real world application on the secondary level as well as on the graduate level, as shown in Luecke’s course. / text