Type of Document 
Dissertation 
Author 
Schellhorn, William

URN 
etd07072005121012 
Title 
Virtual Strings for Closed Curves with Multiple Components and Filamentations for Virtual Links 
Degree 
Doctor of Philosophy (Ph.D.) 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Richard A. Litherland 
Committee Chair 
Frank Neubrander 
Committee Member 
Lawrence Smolinsky 
Committee Member 
Oliver Dasbach 
Committee Member 
Robert Perlis 
Committee Member 
Kemin Zhou 
Dean's Representative 

Keywords 
 filamentations
 closed curves
 virtual strings
 virtual links

Date of Defense 
20050422 
Availability 
unrestricted 
Abstract
The theory of filaments on oriented chord diagrams can be used to detect some nonclassical virtual knots. We extend existing filament techniques to virtual links with more than one component and give examples of virtual links that these techniques can detect as nonclassical. Given a signed Gauss word underlying an oriented chord diagram, we describe how to construct a finite sequence of integers that encodes all of the filament information for the diagram. We also introduce a square array of integers called a MINsquare that summarizes the filament information about all of the signed Gauss words having a given Gauss word shape.
A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. We use the theory of virtual strings to obtain a combinatorial description of closed curves in the 2sphere (and therefore 2dimensional Euclidean space) in terms of Gauss paragraphs and wordwise partitions of their alphabet sets. In addition, we prove that the unordered triple consisting of the Gauss paragraph, the wordwise partition, and a related wordwise partition associated to a closed curve on the 2sphere is a full homeomorphism invariant of the closed curve. We conclude by introducing a multivariable polynomial that is a homotopy invariant of virtual strings with multiple circles.

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