## Electronic Theses and Dissertations

Spring 5-4-2016

#### Level of Access

Campus-Only Thesis

#### Degree Name

Master of Arts (MA)

Mathematics

Robert Franzosa

Andrew Knightly

Nigel Pitt

#### Abstract

A prominent formula in knot and ribbon theory is White's formula that looks at the relationship between linking number, twist, and writhe. After a brief background in knot theory we look at knots, links, and diagrams of each and we examine their associated properties, writhe and linking number. We prove that linking number is an invariant of links and that writhe is not a knot invariant since it is dependent on the diagram from which it is defined. We then extend these concepts to ribbons and diagrams of them. We examine how linking number and writhe are defined for ribbons and we introduce another property, twist. We demonstrate how linking number, writhe, and twist are related in a particular diagram and we examine how the relationship is affected as a diagram changes. The result is the LTW formula for ribbons, a diagram-based version of White's formula. We then introduce Möbius ribbons and show how linking number, writhe, twist, and the LTW formula carry over to that setting. Lastly, we look at the integral-based White's Formula and we discuss how our diagram-based LTW formula relates to it.