Senior Project Advisor

Jeffrey Meier

Document Type

Project

Publication Date

Spring 2021

Keywords

Math, topology, knot theory, braid theory, low dimensional topology, quasipositive braids, ribbon surfaces

Abstract

Meant to serve as an accessible exploration of knot theory for undergraduates and those without much experience in topology, this paper will start by exploring the basics of knot theory and will work through investigating the relationships between knots and surfaces, ending with an analysis of the relationship between quasipositive braids and surfaces in 4-space. We will begin by defining a knot and introducing the ways in which we are able to manipulate them. Following that, we will explore the basics of surfaces, building up to a proof that all surfaces are homeomorphic to a series of disks and bands which have a single boundary component and an introduction to how to view surfaces in higher dimension. After that, we will examine the relationship between knots and surfaces, proving that every knot bounds an orientable surface. We will follow that up with an introduction to braids and a proof that every knot is isotopic to a braid through Alexander's algorithm. Finally, we will dissect quasipositive braids and their special relationship with surfaces in 4-space.

Department

Mathematics

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

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