Title page for ETD etd-07072005-121012


Type of Document Dissertation
Author Schellhorn, William
URN etd-07072005-121012
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 2005-04-22
Availability unrestricted
Abstract
The theory of filaments on oriented chord diagrams can be used to detect some non-classical 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 non-classical. 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 MIN-square 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 2-sphere (and therefore 2-dimensional Euclidean space) in terms of Gauss paragraphs and word-wise partitions of their alphabet sets. In addition, we prove that the unordered triple consisting of the Gauss paragraph, the word-wise partition, and a related word-wise partition associated to a closed curve on the 2-sphere is a full homeomorphism invariant of the closed curve. We conclude by introducing a multi-variable polynomial that is a homotopy invariant of virtual strings with multiple circles.

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