The study of knot invariants is a large and active area of research in the field of knot theory. In the early 1990s, Russian mathematican Victor Vassiliev developed a series of numerical knot invariants, now known as Vassiliev invariants. These invariants have sparked a great deal of interest in the mathematical community, and it is conjectured that, together, they formulate a complete knot invariant. The computation of these invariants is largely algebraic, and unfortunately the values do not appear to describe any intrinsic properties of the knot. In this thesis, a geometric interpretation of the second Vassiliev invariant is provided by examining occurrances of five distinct points on the knot that lie on a common circle in the ambient space. This process is then extended to include an analysis of six-point cocircularities of knots as well. / Graduate

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This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds. / Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded. / The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially. / We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.

Thesis (Ph. D. in Plant Pathology)--Washington State University, May 2008. / Includes bibliographical references.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 2 uppsatser.

Thesis (Ph.D.) -- University of Adelaide, Dept. of Plant Pathology, 1978.

Thesis (Ph. D.)--Ohio University, June, 2000. / Title from PDF t.p.

Thesis (Ph. D.)--University of Hawaii at Manoa, 2003. / Includes bibliographical references.

Thesis (Ph. D.)--University of Hawaii at Manoa, 2003. / Includes bibliographical references.

Thesis (M.S.)--University of Nevada, Reno, 2008. / "May, 2008." Includes bibliographical references (leaves 47-48). Online version available on the World Wide Web.