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.

Invariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be. We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group. We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.

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.

Thesis (Ph. D.)--University of California, Riverside, 2008. / Title from first page of PDF file (viewed Febrary 3, 2010). Available via ProQuest Digital Dissertations. Includes bibliographical references. Also issued in print.

Thesis (M.S.)--West Virginia University, 2002. / Title from document title page. Document formatted into pages; contains viii, 77 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 71-77).

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

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