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 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

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

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

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

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: John Luecke. Vita. Includes bibliographical references. Available also from UMI company.

Thesis (Ph. D.)--University of Oregon, 2004. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-125). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (M.S.)--University of Texas at El Paso, 2009. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

This paper focuses on generalizing unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces. The paper introduces a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. In the research, twisted virtual bikeigebras are used to dene X-colorability for twisted virtual handlebody-links and define an integer-valued invariant of twisted virtual handlebody-links. The paper also includes example computations of the new invariants and use them to distinguish some twisted virtual handlebody-links.

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