Graduation Year

2007

Document Type

Thesis

Degree

M.A.

Degree Granting Department

Mathematics

Major Professor

Masahiko Saito, Ph.D.

Co-Major Professor

Mohamed Elhamdadi, Ph.D.

Committee Member

Mil´e Krajcevski, Ph.D.

Keywords

Fox colorings, diagrams with distinct colors, determinants, links, Jones Polynomial

Abstract

In this paper, we examine Fox colorings of virtual knots, and moves called k-swap moves defined for virtual knot diagrams. The k-swap moves induce a one-to-one correspondence between colorings before and after the move, and can be used to reduce the number of virtual crossings. For the study of colorings, we characterize families of alternating virtual knots to generalize (2, n)-torus knots, alternating pretzel knots, and alternating 2-bridge knots. The k-swap moves are then applied to prove a "virtualization" of the Kauffman-Harary conjecture, originally stated for classical knot diagrams, for the above families of virtual pretzel knot diagrams.

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