Graduate Studies

 

First Advisor

Mark Brittenham

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

Date of this Version

8-2024

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Mark Brittenham

Lincoln, Nebraska, August 2024

Comments

Copyright 2024, Kaitlin R. Tademy. Used by permission

Abstract

A virtual torus knot T(p,q,VC) sits in the intersection of the well-understood torus knot and the not-so-well-understood virtual knot, making it an intriguing object to study.

The unknotting number of a classical knot K is defined unambiguously. However, "the" unknotting number when K is a virtual knot is not as clear to define, since virtual knots have both classical and virtual crossings. We will define virtual unknotting number vu(K) as the minimum number of (classical) crossing changes required to unknot K. Under this definition of virtual unknotting, not all virtual knots can be unknotted. We call those which can be unknotted virtually nullhomotopic.

Working under this same definition of vu(K), Ishikawa and Yanagi establish bounds on vu(K) for a particular family of virtual torus knots VTnpq which they construct, making use of the u-invariant u(K) and P-invariant P(K). We similarly make use of these tools, among others, in pursuit of virtual unknotting information on families of virtual torus knots.

Our findings include families of virtually nullhomotopic virtual torus knots with virtual unknotting number computed or bounded, families of virtual torus knots which are not virtually nullhomotopic, and observations on the relationship between the P-invariant and the size of the virtual torus knot.

Advisor: Mark Brittenham

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