The detection of internal log defects has been shown to have a potential for increasing the lumber value. As an alternative to other available expensive log scanning devices, a method using radio frequency waves has been used to detect the knots. The main focus of the current research is to investigate the effectiveness of using radio frequency waves to detect the knots. Electrostatic finite element analysis is performed to predict the defects in logs. A script has been written using the commercial finite element ANSYS software to predict defects in log sections. The results are then compared with the experimental data measured on actual log sections. Analysis proved that it is possible to detect presence of knots in the log sections.

Berge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1. / text

<p>This thesis is an introduction to virtual knots and the forbidden moves, and the closely related classes of welded and unwelded knots. Extensions of the Jones polynomial and the knot group to the various knot types are considered. We also examine the operation of connected sum for virtual and welded knots, and we review the proof that every virtual knot can be untied using the forbidden moves.</p> / Master of Science (MSc)

In this dissertation, we study the flows on R3 associated with a nonlinear system differential equation introduced by Clark Robinson in [46]. The periodic orbits are modeled by a semi-flow on the L(−1,−1) template. It is known that these are positive knots, but need not have positive braid presentations. Here we prove that they are fibered. We investigate their linking and we construct a zeta-function that counts periodic orbits according to their twisting. This extends work by M. Sullivan in [55], and [57].

No description available.

This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C . The techniques currently available in the literature are either too theoretical, applying to only a small number of knots, or are designed to only deal with a specific knot. The thesis builds on the results of Herald, Kirk and Livingston [HKL10] and Tamulis [Tam02] to give a series of criteria, using twisted Alexander polynomials, for determining whether a knot is of infinite order in C. There are two immediate applications of these theorems. The first is to give the structure of the subgroups of the concordance group C and the algebraic concordance group G generated by the prime knots of 9 or fewer crossings. This should be of practical value to the knot-theoretic community, but more importantly it provides interesting examples of phenomena both in the algebraic and geometric concordance groups. The second application is to find the concordance orders of all prime knots with up to 12 crossings. At the time of writing of this thesis, there are 325 such knots listed as having unknown concordance order. The thesis includes the computation of the orders of all except two of these. In addition to using twisted Alexander polynomials to determine the concordance order of a knot, a theorem of Cochran, Orr and Teichner [COT03] is applied to prove that the n-twisted doubles of the unknot are not slice for n ≠ 0,2. This technique involves analysing the `second-order' invariants of a knot; that is, slice invariants (in this case, signatures) of a set of metabolising curves on a Seifert surface for the knot. The thesis extends the result to provide a set of criteria for the n-twisted double of a general knot K to be slice; that is, of order 0 in C. The structure of the knot concordance group continues to remain a mystery, but the thesis provides a new angle for attacking problems in this field and it provides new evidence for long-standing conjectures.

This thesis is an investigation of the difference between metabolic and hyperbolic objects in a variety of settings and how they interact with cobordism and 'double cobordism', both in the setting of algebraic L-theory and in the context of knot theory. Let A be a commutative Noetherian ring with involution and S be a multiplicative subset. The Witt group of linking forms W(A,S) is defined by setting metabolic linking forms to be 0. This group is well-known for many localisations (A,S) and it is a classical fact that it forms part of a localisation exact sequence, essential to many Witt group calculations. However, much of the deeper 'signature' information of a linking form is invisible in the Witt group. The beginning of the thesis comprises the first general definition and careful investigation of the double Witt group of linking forms DW(A,S), given by the finer equivalence relation of setting hyperbolic linking forms to be 0. The treatment will include invariants, structure theorems and localisation exact sequences for various types of rings and localisations. We also make clear the relationship between the double Witt groups of linking forms over a Laurent polynomial ring and the double Witt group of those forms over the ground ring that are equipped with an automorphism. In particular we prove the isomorphism between the double Witt group of Blanchfield forms and the double Witt group of Seifert forms. In the main innovation of the thesis, we next define chain complex generalisations of the double Witt groups which we call the double L-groups DLn(A,S). In double L-theory, the underlying objects are the symmetric chain complexes of algebraic L-theory but the equivalence relation is now the finer relation of double algebraic cobordism. In the main technical result of the thesis we solve an outstanding problem in this area by deriving a double L-theory localisation exact sequence. This sequence relates the DL-groups of a localisation to both the free L-groups of A and a new group analogous to a 'double' algebraic homology surgery obstruction group of chain complexes over the localisation. We investigate the periodicity of the double L-groups via skew-suspension and surgery 'above and below the middle dimension'. We then reconcile the double L-groups with the double Witt groups, so that we also prove a double Witt group localisation exact sequence. Finally, in a topological application of double Witt and double L-groups, we apply our results to the study of doubly-slice knots. A doubly-slice knot is a knot that is the intersection of an unknotted sphere and a plane. We show that the double knot-cobordism group has a well-defined map to the DL-group of Blanchfield complexes and easily reprove some classical results in this area using our new methods.

Every two-bridge knot or link is characterized by a rational number p/q, and has a fundamental group which has a simple presentation with only two generators and one relator. The relator has a form that gives rise to a formula for the Alexander polynomial of the knot or link in terms of p and q [15]. Every two-bridge knot or link also has a corresponding “up down” graph in terms of p and q. This graph is analyzed combinatorially to prove several properties of the Alexander polynomial. The number of two-bridge knots and links of a given crossing number are also counted.

Mikhail Khovanov developed a bi-graded homology theory for links in the 3-sphere. Khovanov's theory came from a Topological quantum field theory (TQFT) and as such has a geometric interpretation, explored by Dror Bar-Natan. Marta Asaeda, Jozef Przytycki and Adam Sikora extended Khovanov's theory to I-bundles using decorated diagrams. Their theory did not suggest an obvious geometric version since it was not associated to a TQFT. We develop a geometric version of Asaeda, Przytycki and Sikora's theory for links in thickened surfaces. This version leads to two other distinct theories that we also explore.

Celtic art contains mysterious and fascinating aesthetic elements including complex knot work motifs. The problem is that creating and exploring these motifs require substantial human effort. One solution to this problem is to create a process that collaboratively uses interactive and procedural methods within a computer graphic environment. Spline models of Celtic knot work can be interactively modeled and used as input into procedural shaders. Procedural shaders are computer programs that describe surface, light, and volumetric appearances to a renderer. The control points of spline models can be used to drive shading procedures such as the coloring and displacement of surface meshes. The result of this thesis provides both an automated and interactive process that is capable of producing complex interlaced structures such as Celtic knot work within a three-dimensional environment.