Thesis (Ph.D.)--University of Texas at Dallas, 2007. / Includes vita. Includes bibliographical references (leaves 106-108)

In 1984, T. Kobayashi gave a classification of the genus two 3-manifolds with a nontrivial torus decomposition. The intent of this study is to extend this classification to the genus two, torally bounded 3-manifolds with a separating non-trivial torus decomposition. These 3-manifolds are also known as the tunnel-1 generalized satellite knot exteriors. The main result of the study is a full decomposition of the exterior of a tunnel-1 satellite knot in an arbitrary 3-manifold. Several corollaries are drawn from this classification. First, Schubert's 1953 results regarding the existence and uniqueness of a core component for satellite knots in the 3-sphere is extended to tunnel-1 satellite knots in arbitrary 3-manifolds. Second, Morimoto and Sakuma's 1991 classification of tunnel-1 satellite knots in the 3-sphere is extended to a classification of the tunnel-1 satellite knots in lens spaces. Finally, for these knot exteriors, a result of Eudave-Muñoz in 1994 regarding the relative position of tunnels and decomposing tori is recovered.

Ku Wah Kwan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 61-64). / Abstracts in English and Chinese. / Chapter 1 --- Basic 3-Dimensional Contact Geometry --- p.5 / Chapter 1.1 --- Introduction --- p.5 / Chapter 1.2 --- Contact Structure --- p.7 / Chapter 1.3 --- Darboux's Theorem --- p.11 / Chapter 1.4 --- Characteristic Foliation --- p.13 / Chapter 1.5 --- More About S3 with The Standard Contact Structure --- p.16 / Chapter 2 --- Legendrian Knots --- p.18 / Chapter 2.1 --- Basic Definition --- p.18 / Chapter 2.2 --- Front Projection --- p.19 / Chapter 2.3 --- Classical Legendrian Knot Invariants --- p.22 / Chapter 2.3.1 --- Thurston-Bennequin Invariant --- p.22 / Chapter 2.3.2 --- Rotation Number --- p.23 / Chapter 2.4 --- Stabilization --- p.24 / Chapter 3 --- Convex Surface Theory --- p.26 / Chapter 3.1 --- Contact Vector Field --- p.26 / Chapter 3.2 --- Convex Surfaces --- p.29 / Chapter 3.3 --- Flexibility of Characteristic Foliation --- p.34 / Chapter 3.4 --- Bennequin Inequality --- p.36 / Chapter 3.5 --- Bypass --- p.38 / Chapter 3.5.1 --- Modification of Dividing Curves through Bypass --- p.39 / Chapter 3.5.2 --- Relation of Bypass and Stabilizing Disk --- p.40 / Chapter 3.5.3 --- Finding Bypass --- p.40 / Chapter 3.6 --- Tight Contact Structures on Solid Tori --- p.41 / Chapter 4 --- Classification Results --- p.42 / Chapter 4.1 --- Unknot --- p.43 / Chapter 4.2 --- Positive Torus Knot --- p.45 / Chapter 5 --- Transverse Knots --- p.50 / Chapter 5.1 --- Basic Definition --- p.50 / Chapter 5.2 --- Self-linking Number --- p.54 / Chapter 5.3 --- Relation between Transverse Knot and Legendrian Knot --- p.55 / Chapter 5.4 --- Classification of Unknot and Torus Knot --- p.57 / Chapter 6 --- Recent Development --- p.60 / Bibliography --- p.61

Knot theory is a relatively young branch of mathematics, still less than a century old. The development of the Jones polynomial in 1984 led to increased activity in knot theory. Though work is constantly being done in this field, notably the classification of torus knots, double-torus knots are still lacking such a complete understanding. There exists two notations, those of Rick Norwood and of Peter Hill, that describe knots on the double-torus. The ambition of this thesis is to begin to make the case that it is possible to render these two notations interchangeable. Illustrating this will require examining the two notations and finding a way to change one into the other, then check if this process is reversible. If not, then proceed to develop a method that works to convert the second notation back to the first.

Intuitively, a ribbon is a topological and geometric surface that has a fixed width. In the 1960s and 1970s, Calugareanu, White, and Fuller each independently proved a relationship between the geometry and topology of thin ribbons. This result has been applied in mathematical biology when analyzing properties of DNA strands. Although ribbons of small width have been studied extensively, it appears as though little to no research has be completed regarding ribbons of large width. In general, suppose K is a smoothly embedded knot in R3. Given an arclength parametrization of K, denoted by gamma(s), and given a smooth, smoothly-closed, unit vector field u(s) with the property that u'(s) is not equal to 0 for any s in the domain, we may define a ribbon of generalized width r0 associated to gamma and u as the set of all points gamma(s) + ru(s) for all s in the domain and for all r in [0,r0]. These wide ribbons are likely to have self-intersections. In this thesis, we analyze how the knot type of the outer ribbon edge relates to that of the original knot K and the embedded resolutions of the unit vector field u as the width increases indefinitely. If the outer ribbon edge is embedded for large widths, we prove that the knot type of the outer ribbon edge is one of only finitely many possibilities. Furthermore, the possible set of finitely many knot types is completely determinable from u, independent of gamma. However, the particular knot type in general depends on gamma. The occurrence of stabilized knot types for large widths is generic; we show that the set of pairs (gamma, u) for which the outer ribbon edge stabilizes for large widths (as a subset of all such pairs (gamma, u) is open and dense in the C1 topology. Finally, we provide an algorithm for constructing a ribbon of constant generalized width between any two given knot types K1 and K2. We conclude by providing concrete examples.

The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local information and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an extension of planar algebras which are the `right' setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. Using our invariant we can give simple computational proofs of many of these relations, as well as an alternate proof that the MVA and our virtual extension are of finite type.

The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.

The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local information and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an extension of planar algebras which are the `right' setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. Using our invariant we can give simple computational proofs of many of these relations, as well as an alternate proof that the MVA and our virtual extension are of finite type.

Formulas which derive common concordance invariants for satellite knots tend to lose information regarding the axis a of the satellite operation R(a,J). The Alexander polynomial, the Blanchfield linking form, and Casson-Gordon invariants all fail to distinguish concordance classes of satellites obtained by slightly varying the axis. By applying higher-order invariants and using filtrations of the knot concordance group, satellite concordance may be distinguished by determining which term of the derived series of the fundamental group of the knot complement the axes lie. There is less hope when the axes lie in the same term. We introduce new conditions to distinguish these latter classes by considering the axes in higher-order Alexander modules in three situations. In the first case, we find that R(a,J) and R(b,J) are non-concordant when a and b have distinct orders viewed as elements of the classical Alexander module of R. In the second, we show that R(a,J) and R(b,J) may be distinguished when the classical Blanchfield form of a with itself differs from that of b with itself. Ultimately, this allows us to find infinitely many concordance classes of R(-,J) whenever R has nontrivial Alexander polynomial. Finally, we find sufficient conditions to distinguish these satellites when the axes represent equivalent elements of the classical Alexander module by analyzing higher-order Alexander modules and localizations thereof.

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