Understanding the Slice-Ribbon Conjecture Through Algorithmically Ribbon Knots: Inspecting Escapees and Inconclusive Knots

Abstract

A slice knot is a knot that bounds a disk in B4. A ribbon knot also bounds an embedded disk in B4, but with certain irregularities through self intersections. Whether or not all ribbon knots are slice knots is a well known fact among knot theorists. However, the converse is still an open question, and such is called the slice ribbon conjecture. There have been certain cases where this has been proven to be true for certain subsets of knots, such as 2-bridge knots \cite{Lisca} and 3-stranded pretzel knots \cite{Greene/Jabuka}, but not all. In gathering more information about types of knots that are ribbon, Brendan Owens and Frank Swenton developed an algorithm to know whether alternating knots up to 21 crossings bounds a ribbon \cite{Owens/Swenton}. The thesis is divided into three parts 1.) providing an overview of key knot theory and topology concepts related to the research paper at hand 2.) providing a summary and contextualization of Owens' and Swenton's research paper \textit{An Algorithm to Find Ribbon Disks for Alternating Knots} with additional details that would be hopefully accessible to other math undergraduate students 3.) a series of proposals of what could be said about six 16 and one 17 crossing knots whose current status is unknown based on the Algorithm and additional methods. Furthermore, we also inspect some of simplest escapee knots to garner ideas of what could potentially be similar among them. This is particularly because it could give further information on the notion of bifacotorizability that would potentially include escapees.